1. You have to own that domain forever, until or at least until you're 100% confident that an email intended for you will never be sent to that domain ever again. Even then, there are security risks with giving up the domain.
2. You give up some privacy. You can use mailbox aliases but it doesn't really matter if all the mailboxes are tied to a domain registered to your name and address.
2. Whois privacy solves this. Free from any decent registrar.
This issue goes far beyond email alone. The ICANN domain system effectively rents a string out to you on a temporarily basis and mandates that an Impressum be attached to it. It's a deeply flawed scheme when viewed from the context of both historical hacker culture as well as the fundamental values of a free and open society.
As an example Anthropic and OpenAI don't let you change your email address.
Note that I'm not even talking about trying to send email FROM a self-hosted account, but trying to get someone else to send email TO such an account.