But if they violate their promise to Mozilla, won't they be liable? And won't Mozilla have standing to sue?

What do you think damages would be if you violate a contract with Mozilla that puts millions of users at risk? (I wouldn't want to test that)

And won't they get caught during audits? Or at least risk it.

I don't really care about "promises" and contracts made between faceless corporations. I have no reason to trust Cloudflare or modern Mozilla, neither have I a reason to believe Mozilla would litigate against a breach of contract publicly instead of settling privately and secretly to prevent public outrage.

You trust Mozilla code, why not trust them to negotiate on your behalf?

Do you think you'll be able to negotiate a better solution on your own... If so, just change the defaults.

Refusing to host neo-Nazis might be inconsistent, but calling it "evil" is pushing it.

Censorship is evil, regardless of who it is directed against.