An average Russian might go with Mihail for Михаил, yes. The thing is an average Russian doesn't really need to do this kind of transliteration on their own. Maybe when buying things off of an international web-store or something, but when it comes to credit cards, visas, international passports etc, the translitiration is done by people who know better

These people will put Mikhail in there, I assure you. Why? Simply put, it's a common practice right now, the "rules" for it are all but set now. Then, an average Russian will just copy their Latin written name off of those official documents, bank cards when they need to do this themselves.