My question is: if, in any company, your promotion is decided by a committee and is independent from your KPIs, should you leave? Or is this a common mechanism?
It's certainly common for promotion decisions to factor in more than just your KPIs - if anything deciding it purely on the KPI values would be the exception rather than the norm. Even well designed, well managed KPIs are nothing more than good indicators about an employee, they're valuable data when making this sort of decision but they aren't the whole story. So should you leave on that basis? No it's going to be a real slog to find somewhere that doesn't have something functionally similar, and really that's not your issue here - it's your relationship with your old manager that's inhibiting your chances here not the process itself.
Unfortunately it's also not particularly unusual for certain individuals to hold the ability to effectively blackball someone's promotion chances - whether because they have a final approval role or because they carry sufficient weight in any committee-based process. In an ideal world people wouldn't use individual grudges in that way - but it's not an ideal world.
The way I see it you have four options:
Try and repair the relationship
I know it's not an appealing thought given your feelings for your old manager - I don't even know if it's possible. But it sounds as though it's likely to be the most likely path to getting the promotion you seek.
Play the long game
Keep killing it on the KPIs and any other measures of your performance you can - do this long enough and eventually people are going to start seeing your previous manager blackballing you for what it is - a grudge. And if indulging that grudge is harming the business more than promoting you over their objections the business will make it happen whatever the previous manager wants.
Accept remaining at your current level
It's not great as options go - but it is a valid one. If you're otherwise happy where you are then just write the promotion off as something that wasn't to be and carry on as before.
If you're good enough for the promotion-level role at your current company then you're good enough for it elsewhere. Job hunt for roles at that level and when one comes along take it.