Modern war is not total war; the fact that aircraft carriers can be destroyed is mostly irrelevant, because any conflict which escalated to the point where aircraft carriers are being sunk, has a very high risk of also escalating to nuclear armageddon. Modern military operations mostly consist of projecting force into asymmetric conflicts; aircraft carriers do well at this.
Noting that this is correct, but incomplete. They are very important for force projection even in near-peer engagements, since otherwise you likely can't get your planes to where you need them. The question that matters for this is who wins the area-denial / anti-aircraft battle, i.e. can drones and similar get close enough to sink anything, and this is the critical question anyways, since your carriers and planes are useless if you can't get close enough. And this isn't my area, but my very preliminary impression is that AA/AD makes aerial combat fairly defense-dominant.
TL;DR: No, but they are much more vulnerable than they used to be, and the number of nations which can threaten them is increasing because the threats are getting cheaper to deploy.
You have two separate questions at work here, and I think we would benefit from disentangling them. They appear to be:
Considering both of these questions might lead to a natural third question:
Another thing we want to be specific about is: whose aircraft carriers? Most of the answers and possibly also the question implicitly assume the United States, but a US Navy aircraft carrier and anyone else's aircraft carrier are very different propositions.
Onto the first question: vulnerability.
Fundamentally yes; they are large warships and large warships can be sunk. Are they super vulnerable? No; they are very large warships, usually have better signal and detection capability, and have aircraft to use for defense. In the case of US Navy aircraft carriers, they are further surrounded by other large warships, each among the most powerful of their type. Why is this a conversation now and not thirty years ago? Because the stuff used to find and knock out large warships from land is now cheap enough that middle-weight countries can afford them.
Speaking to your specific example:
I.e. can’t an enemy send tons of cheap drone planes and drone submarines to first hunt for its location and then swarm-attack it?
Hypothetically yes. Now the question becomes: launched from where? If these are extremely long range drones, that comes at a premium - this is like electric car batteries, only more so. How to make sure the swarm is sufficiently intact to hurt the aircraft carrier when they get there? If you choose stealth, you have abandoned cheap, which makes a swarm very hard. How efficient are these drones in air-to-air combat against fighters? How vulnerable are they to anti-aircraft guns, and electronic countermeasures? If it is the US Navy again, we are talking about an organization that can put up kill ratios of 12:1 against peer aircraft. A modern carrier wing has ~40 strike fighters. Assuming drones much more capable than we currently deploy seems very reasonable, but 480 of them, plus accounting for other losses, plus the need to actually neutralize the carrier begins to seem like a complex challenge.
Onto the second question: utility.
A short review: aircraft carriers carry aircraft to sea. Their job is to bring these aircraft within range of the objective, and then the aircraft do the work they normally do (patrols, reconnaissance, stopping enemy fighters, bombing stuff). Then they can move to another objective.
Airpower is extremely useful, and control of the air makes everything easier for your side and everything harder for the other side, on land or sea. What does a better, or at least more efficient, job of providing airpower in naval engagements? The short version is that there are no replacements currently, and developing replacements introduces a series of tradeoffs and a very high cost.
Onto the final question: worth it?
There isn't, so far as I am aware, any plan which can replace the capabilities provided by aircraft carriers to the United States. This means we have to continue to pay, or go without. Another item to consider which is specific to the United States: the existence of our Carrier Groups is the bedrock assumption upon which the naval balance of power is calculated. This becomes significant when considering this question for other navies, which lack the experience, production support, strategic integration, and deep other fleet strengths.
My current opinion is probably yes for the United States, and probably no for all others.
Predictably, the problem is complex and there are many more dimensions to consider than I have presented, all in deeper detail. If you want to get a sense of how the discussion looks within the defense community, consider these sources:
Defense One, a popular defense news site.
War on the Rocks, long-form and more opinionated.
RAND, the military think tank par excellence.
First principles reasoning says the answer is yes, for the reasons you mention--in a fight against a peer adversary at least. But maybe there are some considerations I'm not aware of. I'm not a domain expert.
Carriers are still super valuable for modern war against non-peer adversaries. When the USA fights Iraq, Afghanistan, Syria, etc. the carriers are well worth their cost, I would bet.
You may be interested in Bean (Naval Gazing)'s essays on whether the carriers are doomed. I don't find them convincing myself but I do recognize he knows way more about this than me.
EDIT: Also, in a war against a peer adversary carriers would still be useful if they stayed out of range of enemy missile and plane swarms. There's a whole globe to fight over, after all. Carriers could project power to the regions that are far away from the homelands of both nations.
Maybe, but very likely still useful. They are vulnerable if they can be directly attacked, but largely in the same ways and for the same reasons as an airfield. Same as for ground based airfields, if fighting is happening around and on top of the carrier things are already going very very badly. Doctorine generally centers on not letting anything hostile get close enough to take a shot at all, and so carriers can be found in the centers of fleets that are spread out to screen the carrier (in addition to the carrier's own air patrols).
As already mentioned, there's an arms race with respect to cruise missiles; missiles have become extremely capable in terms of range and finding a target, and extremely accurate. If one hits it's probably sufficent to sink a ship regardless of size. Point defence seems to be lagging slightly at present, but it seems premature to call the race. Public information on exactly where an arms race like this is usually lags 10-20 years, but I'd consider the success of anti-satilite and anti-balistic-missile programs strong evidence that shooting missiles with other missiles has become reality.
As for drones, keep in mind that cruise missles of the type that can sink a large ship in one hit mass tens of tons; likewise for torpedos. While drones could certainly be a threat, it's unclear they'd represent an asymetric threat (e.g. the carrier could presumably also launch drones too and so respond in kind). Submarines are very much an asymetric threat to carriers, but that's not new, and I'd be skeptical that drones massively alter the balance of power there, given the extreme difficulty communicating with anything underwater - subs require significantly more autonomy than UAVs.
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To destroy an aircraft carrier you must first find it and in a war the US would prioritize taking out the enemy's ability to locate our aircraft carriers. Since the carriers move, knowing were one was an hour ago might not be enough information to destroy the carrier. In the next future aircraft carriers might be protected by laser anti-missile systems that could handle having only two seconds to destroy multiple incoming missiles.
Not an expert, but they seem to be less useful in a conflict with a nuclear power that possesses something like this: https://en.wikipedia.org/wiki/Kh-47M2_Kinzhal
It seems like aircraft carriers are a good candidate for something that only exists because it made sense in WWII and it looks the part of “impressive military asset”, i.e. it’s all larping at this point. It seems vulnerable to attack relative to its huge cost because offense has an advantage over defense: I.e. can’t an enemy send tons of cheap drone planes and drone submarines to first hunt for its location and then swarm-attack it?
Note: I don’t know anything about this subject. The ideal answerer is someone with domain knowledge who has good epistemology; that’s why I wasn’t satisfied with a Google search and I’m asking here.