Not quite what I'm looking for, but thank you for the Hero mention!
A number of games use action phase tables with one action per phase, but various characters having different numbers of actions. Not quite an AP system, but a conceptual step just beside it.
Hero System is the most obvious.
the old GW
Judge Dredd was another - you got one action per 10 initiaitive (an attribute), scattered over the 10 phase turn. If A had 2 actions and B had three, due to the table, they'd wind up B→A→B→A→B...
WFRP 1E: Actions was an attribute; you got your first action on your Initiaitive Att's point (modified for weapon/action); if you had 2 actions, the second was on half that initiative; if you had 3 actions, they were on full, 2/3, and 1/3... Only one of them was allowed a move, IIRC...
Some board games with noteworthy
Star Fleet Battles: Noted now for the 32 impulse game; early editions you got to use a shorter chart - Designer's Edition has 10 and 20 impulse charts. Your speed determines which impulses you move in. Using the 10 impulse chart, Speed 3 moves on impulses 4, 7, and 10, while speed 6 moves on impulses 2, 4, 5, 7, 9, 10, and speed 5 on impulses 2, 4, 6, 8, 10. Note that you have firing opportunities on EVERY impulse, but no weapon can fire twice in a turn nor within a quarter of a turn of firing in a prior turn...
Commander's Edition also had 8 and 16 impulse charts... better fits for the "quarter turn rule"... and that QTR affects a bunch in play. Not just weapon fire, but a bunch of other "systems" have QTR limits.
Starfire also uses impulses, but handles speeds differently: you move in impulses 1 to speed... assuming you want to - speed is a fixed number. Unrealistic, but closer to an AP system where you have to decide whether to use one to move, turn, both, or neither each impulse. Firing is a whole separate step from impulse procedure.