Software is just an idea written down rigorously. Various societies created various conventions and social contracts to control dissemination and usage of ideas, both in their pure and written down forms. Capitalist societies generally defer to the author of the idea for how they want it handled (at least for the first few decades), so that the author can earn some money from it (of course, even ideas are monetized under capitalism) - this is patent and copyright law.

The free software movement is just a novel application of the copyright law. By sharing ideas freely but with a license that forces everyone using the idea to share their derivative ideas freely as well, it is attempting to destroy the spirit of copyright law by using the letter of copyright law.

With all this in mind, let's examine what it would mean to add the "don't be evil" clause to an otherwise FOSS license.

In ideal circumstances, a society's system of laws and social norms should incorporate "don't be evil" in it already. Hate speech and nazism should be prohibited and punished, so the clause would be superfluous.
In "ordinary circumstances" of neoliberal capitalism, there are agencies that will be acting in bad faith but will stand above any laws, be it geneva conventions, hate speech laws or (boring) copyright law. You won't be able to enforce a "don't be evil" clause against the CIA or ICE or the Rockefeller. They can just take your software and use it, so the clause would be of little use typically. This is partially applicable to our current situation.
In extraordinary circumstances, such as capitalism in advanced decay a.k.a fascism, the law will be ignored by most evil actors anyways. Law is just a social contract and fascism is deliberately breaking all social contracts. Nobody will enforce copyright law in favor of an individual FOSS developer, especially against someone who's on the side of the regime. So the clause is completely useless. This is also applicable to our situation.

There is some edge-cases in the middle where a "don't be evil" clause might make a bit of sense. If the contract law (which includes copyright law) is still well-respected, but the social contract itself is falling apart around it, it might be used to prevent some nazis somewhere from using your software for a short while, but that situation is always unstable and does not last. In any case nazis are known for ignoring all social contracts, including court orders, so even this is questionable.

There are also downsides in any "don't be evil" clause, because it requires you to rigorously define what you mean by "evil". This is actually really hard to do well without relying on existing laws (which ruins the point), and will usually either leave nazis leeway to get away with using it, or harm legitimate users, or both - especially because legitimate users are less likely to try pushing the boundaries.

This is explicitly different from what Bluesky is doing. They are hosting known nazis. Nothing is stopping them from banning ICE and making it into a point of pride, it is really easy. There is no downside, no legitimate user hurt. It's as easy of a decision as one can make.

To reiterate,

So the mastodon service supports Nazis.

Mastodon-the-service doesn't really exist (unless you count mastodon.social). But the fediverse in general is not supporting nazis. Nazis are banned and defederated.

Mastodon-the-software may "support" nazis in the same way as the idea of a printing press (from your other comment) supported nazis.

They’re providing software to Nazis, and I don’t really see how that makes them better than providing a place to post.

Bluesky is categorically worse because it doesn't have the "don't be evil" clause in the software licenses either, and it is hosting nazis directly on the platform they run.