If you own the domain being used, I assume you also host your own email... You can't just make a new address for this and have them all forwarded to your actual email?
"This_is_not_generic" @ "your actual name"
Unless they block that too, I don't think they're trying to force those services on you; they're just popular options and this is an automated response sent by an automated process that only checks the first half of the email and not the domain.
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.