The basic question I'm asking is the following:
Suppose you didn't know humans exist. Could you still deduce that general intelligence is possible, and if so how?
This may be a weird hypothetical to imagine, so here's my best attempt to give a story to aid the counterfactual imagination:
At some point, you gain the ability to observe an alternate universe, which you eventually call the "Earth-universe". The fundamental laws of physics are very different here, and you get to work studying it. At first it feels very weird and you have almost no way of making sense of the raw sensory data. Eventually you learn that there are these things "galaxies" and "stars", and it takes you thousands of years to derive Newtonian mechanics. At this point, you've finally gotten your bearings, your intuitions aren't so horrible anymore, and you can go on to develop general relativity, quantum mechanics, etc. But this is still slow [you're approximately as smart as humans].
However, there aren't any humans in this universe, or any intelligent life. You just focused your sensors on this planet "Earth", where there are some really interesting chemical reactions that perpetuate themselves. You've figured out a lot about paleontology and evolution by natural selection and can see the progression of complicated life forms that have evolved on this planet. Apparently, there used to be these things called dinosaurs, but--just your luck--they died out a couple million years ago, but it's interesting to see how the remaining life's been evolving into the new niches...
Something you wonder to yourself:
Is intelligent life possible in this universe? Not necessarily on this planet, but in general? Like, if it's going to show up anywhere, it would probably be here on Earth (at least of the places I've looked at). But even if it doesn't show up here, that could just be a contingent fact about how things have played out. Does the physics/chemistry of this universe allow for some mind to be intelligent, i.e. to have a similar ability to "figure out" its surroundings much in the way I've been figuring out this universe and my own.
Of course you trivially know that intelligence is possible in your own universe, but you figured out a long time ago that nothing you knew about your universe was at all useful for deducing anything about this one. You tried using your old intuitions, you tried for the first thousand years, but ultimately you just had to scrap all those intuitions and start from scratch. Now, you don't even know how to reason about your own universe anymore...
Note also that you're ways of observing/interacting with this universe are limited, but you can interact with the world enough to run experiments. However, you're own technology isn't much more than what humans actually have, in particular you don't have infinite compute, not even enough to run a full-scale simulation of Earth.
Optional Assumption ("Easy Mode"): You've been running a lab in this universe where you've been trying to build "intelligence" yourself. You've tried all sorts of things, from selectively breeding some of the seemingly "more intelligent" animals to building some silicon-based devices you call "computers" and getting them to do some very specialized intellectual tasks. Lately, you've had some success with a particular approach to the latter, where you've been designing and training "deep learning" algorithms that can e.g. beat you at certain abstract strategy games.
Optional Assumption ("Hard Mode"): The math of the Earth-universe is also much different from what you're used to (not just the physics). For instance, in your own universe the "Turing machine" was not a coherent concept, and you had to figure that stuff out along with the physics.
ETA: "General intelligence" := "human-level intelligence" := "the intelligence you have in this counterfactual", i.e. you're wondering if it's possible for a being to exist that can understand the Earth-universe as well as you have, and even (more slowly) figure out your universe if given access to it. Here's some possible ways one can answer this question [rot13'd to avoid spoiling people's answers]:
1. Nethr gung aba-cevzngr navzny vagryyvtrapr nyernql trgf hf "zbfg bs gur jnl gurer", naq tvira gur nccebcevngr raivebazrag, vg fubhyq or cbffvoyr va cevapvcyr sbe n fhpprffvba bs navzny fcrpvrf gb ribyir gur erznvavat pncnovyvgvrf.
2. [Rnfl Zbqr] Nethr gung qrrc yrneavat nyernql trgf hf "zbfg bs gur jnl gurer", naq vg fubhyq or cbffvoyr, jvgu rabhtu genvavat qngn naq gur nccebcevngr nytbevguzvp gjrnxf, gb trg n qrrc yrneavat nytbevguz gb qb trareny-checbfr ernfbavat engure rssvpvragyl.
3. [Uneq Zbqr] Nethr gung uvtu-yriry zngurzngvpny pbafgehpgf, fhpu nf havirefny Ghevat znpuvarf (be creuncf zber pbaivapvatyl, NVKV), fubj gung yrneavat va irel trareny raivebazragf vf cbffvoyr jvgu hayvzvgrq pbzchgr. Gura nethr gung zhpu bs guvf pna or nccebkvzngrq ol srnfvoyr (r.t. cbylabzvny gvzr) nytbevguzf gung pna eha va erny-gvzr ba n oenva/pbzchgre zhpu fznyyre guna n cynarg.
This still leaves the question of why the chemical reactions on other planets haven't begun colonizing the galaxy, since it seems likely that the chemical reactions on Earth will (eventually) do so.