Yes. The politician wins. Focus groups and polls are clear on this, they won't vote for random people. Trump would also beat them easily.

(Obviously some probability you get someone effectively not random from a random draw, but it's very low.)

Random person most likely has an IQ of 100 which is a standard deviation or two below a random president. Random person most likely has less talent and experience at politics, promotion and attack than a politician. Random person is most likely less physically attractive, less charismatic, and less wealthy than a politician. Random person doesn't have a gigantic support apparatus behind them (although even if they do they're probably still screwed because it's not enough to make up for the rest of the deficiencies). As others say it won't even be close. Random person will predictably be slaughtered. 

Not a very principled answer, but: 98%

There is already a substantial preference for incumbents (about 65-35 I think), and I think this would be much stronger if the challenger was completely unaffiliated with politics (I want to say something like 90-10 if challenger and sitting president were equally electable otherwise, maybe 75-25 if the challenger is actually substantially better than the president, 99-1 if they're just average).

Say there's 5% chance they're equally or more electable, 2% they're substantially more electable. Then there's 0.5% on them being equally electable and winning, 0.5% on them being substantially more electable and winning and 1% on them being less electable but winning anyway.

So 98% overall.

Edit: didn't consider that the major party might organise in support of the candidate, this changes things a lot. Say there's a 20% chance that the out-of-power party campaigns for the rando (40% the rando is somewhat-aligned, 50% conditional that the party can get their shit together to campaign for them); then the incumbency advantage will be more like the standard 65-35, assuming equal presidentiality (but I still think they're probably a lot less presidential)

All told...this gets me to 96% chance the sitting president wins.

To add to the consensus— if a random person actually had a favorable matchup against a career politician of any stripe, that would be a massive low-hanging fruit for existing political actors to capitalize on. The RNC and DNC would be falling all over themselves to present "average joe" candidates if doing so provided a consistent advantage. They're not, so it follows that either those organizations are both highly un-optimized for winning elections (probably not; too much money at stake) or else that the evidence doesn't bear out that they should.

Politicians are good at campaining. Most people are not. A moderately successful politician at the national level would shred through any random people.

So, the three big factors:

• What part of the general public is completely unelectable against anyone stable enough to be a professional politician^[1]? I don't believe a majority of the general public is willing to put, for instance, a convicted murderer, or an unemployed alcoholic in the White House even if they dislike the sitting candidate. Most voters are older, and older people like stability.
  • I don't think I can confidently estimate a number, here. I'll note that intelligence isn't the core factor; I can certainly see either party putting a few handlers around a very low-intelligence person as long as they're manageable, and blaming various -ism's whenever someone claims their candidate isn't smart enough for the job.
• What percent of the general public is more appealing to the opposition party's core power blocs (or, perhaps, blocs within the ruling party?) than the sitting president? For example, the average citizen is much less amenable to playing gently with Wall Street, the Chamber of Commerce, powerful unions, the defense lobby, or a variety of foreign countries than the average politician of either party is. One can imagine that the average GOP donor might be better served by a DNC president than by the election of a random Republican voter, and vice versa for the Democrats.
  • We saw a significant portion of the neoconservative (pro-immigration, pro-Israel/interventionism) sphere pivot to the Democrats during Trump 1, which was much less amenable to their interests than the usual Republican presidency^[2], but it ultimately didn't prevent his election. Still Trump was a famous businessmen with plenty of connections, and thus a known quantity to some degree. Nonetheless, Trump was massively outspent in 2016, and had to win despite that.
  • Voter blocs are also significant here, of course. A proven anti-war candidate could get crossover vote from the majority of Republican under-60's who are souring on Israel. Or, if our random was a Republican, a firm anti-H1B stance would be pretty popular among young Americans of both parties, especially if the candidate was levelheaded enough to frame it as a labor rights issue when presenting to liberal voters. There are many issues that are popular among the public but unpopular in D.C., which I think is the crux of what OP is getting at with the question
• The current president himself is the last factor, and I think it's a relatively minor one. 2016 Trump could probably thrash an unremarkable random citizen who passes the first two checks, while most other presidents would have more of a fight on their hands, simply because the 'outsider' vote would be contested rather than given outright. Besides that, a relatively put-together candidate like Romney or Obama could easily win older status-quo voters when pit against an ordinary person with a less manicured public persona, whereas more abrasive candidates like Hillary or Kamala, or candidates like Biden whose faculties were in question among many people, including many Democrats.

Doing my best to work with these fairly hard-to-quantify constraints:

• LLMs are pretty decent at aggregating long lists of broad categories and guessing at correlations, so I threw the "what share of the public is fundamentally not electable" question at LLM Arena until I got an answer I consider reasonably strong.
  • Age is a big issue, and a lot of people are over age 80. OP mentions 'eligible' in his question, so I'll assume that being too young isn't a factor here.
  • Deceased candidates have won posthumously, and candidates who never campaigned at all have received fairly high vote shares in elections so long as they've kept the nomination.
  • Overall, the higher end of estimates has been 5%, and my initial hypothesis that even mediocre (as it turns out, even deceased!) people can do fine in elections so long as they don't get in the way of their handlers seems well-supported.
• As far as power blocs go, I'll consider the current president. A lot of the GOP establishment resents him, but at the same time, he hasn't gone as hard against them as he could have, and has compromised with them when he didn't have to. A lot of their donor class has done well under him, and this class overlaps significantly with that of the DNC. In 2024, there was a general sense that the political and financial establishment had come to terms with Trump, and weren't as upset with his candidacy as they were in 2016, though notably he was still outspent by a substantial margin: President Trump spent approximately 360 million USD, whereas Harris spent more than 800 million USD.
  • Trump has declined in support among his base, but he has still fairly reliably been able to fight out primaries that he cares about, and most Republicans would probably still not want to run against him. Moreover, while Democrats don't like Trump, they've fairly reliably aligned with more establishment-friendly Republicans against the more populist candidates who would comprise a majority of willing Republican opponents. The "never trump" movement gets a lot of press in liberal circles, but is effectively nonexistent among the general public, and would have negligible support among Republican voters while likewise alienating many Democrats with their broadly neoconservative foreign policy preferences.
  • Among Democrats, I think Kamala Harris is a pretty effective model of how the median non-politician Democrat would perform in an election against a Republican president. The party establishment embraced her, she was provided with a relatively friendly media ecosystem, and while she had many gaffes that hurt her candidacy, there was no big, shocking problem that killed her campaign. Nonetheless, she didn't have the charisma expected of a politician, nor the personality needed to work effectively with her staff^[3].
  • Independents usually, ultimately fall along party lines in politics, but have one or more issues that preclude their association with major parties. While I don't think that this means that an Independent would necessarily be a polarizing crank, I do think it precludes outreach to the full slate of interests that make up a major party, meaning they'd be weaker than a generic candidate by a significant margin.
  • I'll note that many people are apolitical. For the purposes of this analysis, for the reasons discussed above, I'll assume that the people around them (friends, relatives, other loved ones) will move them into one of these three camps at roughly equal ratios to those seen in the political population.

Putting it all together, 22 percent of Americans are 80 or older, ~5 percent have something along the lines of a disqualifying criminal record, mental disability, or social media presence, roughly one third are same-party and would either refuse to oppose their own president or do a shoddy job of it, and one more third are independents who would struggle to make the right compromises to present a strong candidacy. This gives us a baseline of 24 percent. Polymarket odds in the last days of 2024 gave Trump a ~66 percent chance of winning, which, if we assume that the lack of Harris's political experience and Trump's relatively less enthusiastic support cancel out, gives us an estimated 8 percent chance that a random U.S. citizen would unseat President Trump in a snap election.

1. ^{^}

   Yes, yes, I know, low bar, especially nowadays. But no matter how loony some congressmen are, they had to build coalitions and manage donors. There is a bar there, and completely non-functional people wouldn't be able to clear it.

2. ^{^}

   It's hard to remember this given the Iran War, which has lost him many of his noninterventionist supporters, but Trump 1's foreign policy was characterized by some degree of de-escalation with e.g. Russia and North Korea, and a steadfast refusal to expand operations in Syria as much as many GOP and Democratic politicians wanted him to.

3. ^{^}

   Reading between the lines in Harris's 2024 election memoir has been sobering for a great many Democrats and deeply entertaining for quite a few Republicans.

The VAST majority of adult citizens would decline rather than suffering the scrutiny of the public, and the unpleasant work of internalizing and repeating the lies needed to win, especially with the expectation that it would get even worse if they actually won.

General Sherman had this one right.  

There's enough magic/counterfactual assumptions in the question that I don't think I can provide a probability - it's a different universe than I live in, so presumably has different voters and different populace.  

Current president of Poland went from being relatively unknown parliament member (I'd bet, less than 5% of general public knew who he was? Also there's another public figure with the same last name, so maybe even some people who knew the name were confusing him with the other guy) to being the president within one year. So with longer timescale, big support from one of leading political powers, and some political backlog it is possible. There are surely similar stories around the world, of outsiders rising to leaders. Of course what you're asking is much more extreme, but that provides some staring datapoint.