I think there’s nuances to this. I’ll throw some of my thoughts out:

It seems that satisfaction living in captivity varies significantly by species (even between individuals). Many reptiles may be fine with a relatively small habitat while animals with large ranges in the wild (eagles, elephants, cetaceans, etc.) require far more (I suspect this would be prohibitively expensive).

All of this is an opportunity cost: can we do more good with the money? Arguably the greatest amount of suffering is in the smallest animals (insects, aquatic invertebrates like shrimp) because there are so many of them (r-selected species). We can’t practically really take care of them (typically short maximum lifespans, difficulty or impossibility of medical treatment, etc.). Magnus Vinding has proposed wildlife antinatalism - that we should prevent wild animals, especially small ones, from being born in the first place. Similarly, Brian Tomasik has argued that habitat loss/deforestation is one of the best ways to reduce WAS, at least in the short to moderate term. It reduces the base of the trophic pyramid - producers like plants - that insects feed on. Less food -> less survival and reproduction -> less insects born -> less (net) suffering. Alternatively, especially at this early stage of WAS having social importance, we could fund outreach/communication or research.

Historically zoos used strong conservation messaging, which likely perpetuates WAS (and an idyllic view of nature). It would take a lot to undo this. Additionally, zoos receive millions in funding from governments and NPOs for conservation. This may include breeding animals, which we shouldn’t do.

I’m mixed on carnivores. Of course we don’t want to support factory farming. It would be best to feed them, for example, cultured meat, but that’s not economical yet. Despite this, it might still be less suffering than had they lived in the wild. Just comparing the deaths, I would think that terrestrial farmed animals are, on average, somewhat better off than being eaten alive because most countries have some level of stunning laws. With farmed or wild caught fish I’m less convinced (lacking of stunning means they can still be conscious for 2+ hours after being caught, comparable or worse to being eaten alive?). In general I think eating larger farmed animals contribute to less suffering per unit of protein, so beef would be the best (not accounting for emissions). Chickens, fish, and especially shrimp and insects could increase suffering. Again, it’s hard to say for sure. Lastly, if animals in captivity live longer (from what I saw it’s ~2x long for lions), that’s more animals killed to feed them. This may be reduced by their lower energy needs. Lastly, per Tomasik, it’s less clear that removing animals from the top of the trophic pyramid

Regarding what nature could look like for wild animals, you may be interested in David Pearce’s work) or Herbivorize Predators. The ideal to me would be no beings capable of suffering that provide ecosystem services humans currently need - so only plants, fungi, microbes, etc. - although I think this is unrealistic.

I’ll end it here for now. As you can see, there’s a lot of unknowns. In the short to medium term, I would lean towards this not being our most effective strategy. As mentioned above, outreach/communication/building community, research, and/or donating to effective charities (in no particular order: Animal Ethics, Center for Reducing Suffering, Rethink Priorities). Very few people, even vegans, care about WAS. I’d recommend much of Magnus Vinding’s work. While I haven’t finished it, I’m a big fan so far of his book, Suffering Focused Ethics (free link from his website) which includes some practical takeaways. I also want to read his book, Reasoned Politics.

Personally I think in almost all cases animals have better lives in zoos than the same species has in the wild, if the zoos are well-funded and make some effort to be "humane". Unlike wild animals, they will get medical treatment, and won't have to worry about predators, parasites, or starvation (orcas MIGHT be an exception because they seem to do particularly poorly in captivity and (adults at least) have no non-human predators in the wild).

However that really doesn't have much at all to do with zoos because the animals in them are bred for the specific purpose of being displayed. They wouldn't even exist in the first place if not for zoos. I am opposed to zoos for that reason, not because I think they'd be better off in the wild; these beings shouldn't be forced to exist just to be a spectacle for humans. For similar reasons I am also opposed to exotic pets, and would only ever adopt pets or take in stray cats.

High quality animal sanctuaries are good though

And ideally we should have a lot more control over nature

Zoos are an incredibly powerful tool for ecological outreach.

Ecological outreach is necessary so that we don't burn down the planet and kill ourselves while trying to climb the tech tree.

Climbing the tech tree is necessary to reach a position from which we can put life on Earth out of its misery.

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Zoos are a necessary evil, and quite honestly not that big of one in most cases - I hate the existence of zoos far less than I hate the existence of wildlife, simply because it entails less total suffering and less per capita suffering.

Is there any real evidence that it's the case?

They die earlier in the wild in almost all cases (and in much more violent fashion), carry greater parasite loads, are wounded more frequently, and generally tend to be less healthy overall. I think that counts as evidence enough that they have lower quality of life.