Domain registration information can usually be found out somehow, although these days you have to jump through some additional hoops to get it, and those hoops are designed to discourage automated lookups. The privacy gains you get from hosting your own email server, though, are massive and IMHO more than worth it. If you are not hosting your own mail server, then the most you can expect from having your own domain is nicer looking email addresses. Depending on what your hosting provider supports you might also get unlimited aliases, maybe even regex aliases, which can be very helpful when handing out mail addresses to various companies and internet services.

If your main concern is that your email address should not be associated with your real identity, your best bet is to just use a VPN to connect to any large email hoster, like ProtonMail. (Obviously don't use Proton Mail if Proton is also your VPN.)