Your own domain is not great for privacy though like others have said the registrar can hide your info at least from whois. If you already have a domain lookup the whois record and see what it says. Presumably even with whois privacy your identity is probably discoverable.
Custom domains are not great for deliverability too. Though mostly mine is fine. Sometimes Yahoo and ATT manged accounts give me delivery issues.
What your own domain is good for is nice, long term, and portable addresses. Also for many cheap addresses. I get something like 30 email accounts with my basic Namecheap cPanel account for about $25 per year.
Yes. I do not use them all, but I can and my cPanel shared hosting only costs $25 per year. I can use web hosting part too if I want. It is all included. The above cost does not include the domain name itself.
See: https://www.namecheap.com/hosting/shared/
Edit: Looks like after first year it will be about $50/year.