I can't speak for Foundation, but I do know that - especially with outgoing emails - self-hosting these days is a pretty big challenge.
Some providers will immediately flag your emails or delete them outright due to too strict firewall settings. Imagine a customer not receiving a response because their email provider didn't let Foundation team's reply through. That will probably be the main issue.
Sure it can happen, but every serious company in the world have their own email with domains, it would be silly if everyone would use just gmail.
Blocking usually happens if people are using shared hosting, and I am not 100% sure but I think that ledger and trezor considered switching to self-hosted emails after leaking of their newsletter third party partners.