And thus you made me reinstal BG3.
Between Pathfinder, Solasta and BG3 I might be mixing things up but I am pretty sure spells like hold person, charm person, slowed or Tasha's Hideaus Laughter should make a saving throw every turn. Should is an operative word though.
Edit: I can confirm that Tasha's HL and hold person makes ST every turn. Charm doesn't, didn't try slow.
But now that I think about it... What about flying creatures that doesn't need feets (nor wings for some of them) to move ?
Shouldn't they be affected by frozen?
Ugh, that's a good question. If BG3 had a proper flying mechanics (like Solasta) I might suggest to ground flying enemies (bring them to ground level) when frozen.
Biggest benefit of flying in BG3 seems to be an ability to ignore surfaces, but with change from surfaces to status effects, I don't think making flying enemies ignore "frozen" effect (or burning) makes sense. I don't think it would feel wrong to freeze "flying" enemies in place, as with how they are presented in BG3 they feel pretty grounded anyway. It does feel like changing surfaces to status effect, could negatively impact flying though, as it's expensive spell that would bring even less value to combat.
edit:
Reducing character's movement range, would make things simpler, but as you mentioned hamstring shot already fulfills that function. Couple options I can think of:
1) grounded or flying - a unit can't move nonetheless
2) a character can't move, but can fly freely (legs are frozen, but he isn't frozen to the ground) - makes more senese with enemies with legs, one can wonder what is being frozen for enemy like Observer - though I don't know how the thing flies to begin with
3) flying range is reduced when frozen.
4) flyinbg creatures when frozen loose their flying status, but can still walk (or crawl - or not if they have nothing to walk or crawl with) - could walking spead be creature specific?