It’s pretty common to own a domain but not actually host the email server; doing on-premises email is a security PITA and most providers simply blacklist large swathes of residential and leasable (e.g. VPS) IPs.
Unfortunately, if you get someone else to host your email, they often charge by the account, not by the domain. Setting up a new mailbox is therefore irritatingly expensive.
A catch-all email works well, though, and is free from most of the hosting providers. Downside is you get spam…
Jane@JaneDoe certainly seems more common than mail@JaneDoe.