One thing I really don't like is that the teachers are House-identified. They're players in the game, and it's OK for them to arbitrarily punish kids from other Houses and show favoritism to their own. That's like making coaches the referees.

It's inevitable when you recruit all your teachers from the school alumni, which itself is more or less inevitable when you're the only school in the nation.

I suppose you could rule that upon taking the job each teacher gets assigned to a new House at random other than the one they were students in (note that this would be a purely informal role, except for the four Heads), but I doubt it would be very effective and not counterproductive.