I tried both hosting my own mail server and using a paid mail hosting with my own domain and I advise against the former.
The reason not to roll out your own mail server is that your email might go to spam at many many common mail services. Servers and domains that don’t usually send out big amount of email are considered suspicious by spam filters and the process of letting other mail servers know that they are there by sending out emails is called warming them up. It’s hard and it takes time… Also, why would you think you can do hosting better than a professional that is paid for that? Let someone else handle that.
With your own domain you are also not bound to one provider - you can change both domain registrar and your email hosting later without changing your email address.
Also, avoid using something too unusual. I went with firstname@lastname.email cause I thought it couldn’t be simpler than that. Bad idea… and I can’t count how many times people send mail to a wrong address because such tld is unfamiliar. I get told by web forms regularly that my email is not a valid address and even people that got my email written on a piece of paper have replaced the .email with .gmail.com cause “that couldn’t be right”…
All good advice. I’d recommended protonmail for mail hosting - got very good experience with them and the onky downside is you have to use their client.