All my medical places I use do that, with the note that you can also use their app. Good system.

Unfortunately, my doctor's office texts me their bank account number saying "please pay $75 to this account". It told them that's putting people at risk of phishing but they didn't care.

Personally, I'd rather they put the HIPAA message content straight into the email, and let Gmail sort out the priority. About 90% "you have received a message" notifications are not actionable: "you made an appointment" or "take this survey nobody cares about."