I know at least two services that offer hosting with your own domain: https://my.snikket.org/ and https://account.conversations.im/domain/ so I suppose it is not that complex to setup.

Correct, but that means you cannot share that domain securely with, let's say, a website. No ?

TBH I am not well-versed into DNS and certificates and security, but I guess you have to trust your provider with a certificate for `example.com` if you allow them to manage JIDs of the form `anything@example.com`, indeed. Again, I am not very knowledgeable about the subject, so take this with a grain of salt.