Solmyr, Demeer, and Aislin are cheaters (and/or) kamikazes; do the rest have a right to exist?

I'm talking about the fact that in a random start, mages might not even appear in the game.

Well, don't confuse a tournament with seasonal rules!
For a tournament, if testing is needed, it would indeed be desirable to require playing only with these heroes (specialists in 4th-level magic).
But for a season, a random start is better, and after the experiment, either the ban on mages will be kept or lifted.
P.S. The topic, although it suggests banning/unbanning specialists in 4th-level magic, if you read carefully, you would also see a suggestion to make the starting hero random... both of these issues are independent.