What matters most of all is a win condition. If a battle has 2 sides & 1 of them has no way to win, it's not really a "battle" anymore, is it?
Some hax matter more than others.
Having something like Transduality, certain types of Acausality, Non-Existent or Abstract Physiology, or Noncorporeality or Elemental Intangibility or High Enough Regeneration can make you nigh impossible for certain foes to kill.
It's rare, but some kind of range can matter a lot. If you can surround yourself with AoE, or attacks, &/or hit your opponent from out of range. Sometimes this includes like from extremes such as other universes, multiverses, dimensions, etc.
However, for that kind of Range to be relevant, you need the AP, Striking Strength or Lifting Strength or Hax to actually use it.
Speed is often equalized. Against a fast enough opponent, they can take action & win before you even make a move. Or take action fast enough that they can advance their own win conditions while denying your own. Making counter-attacks, moving in &/or out of range, etc.
When you skew the action economy, things predictably get unbalanced fast. Hence speed equalization being a thing.
AP matters if you have the means, opportunity, knowledge & volition to use it. Being able to instakill your opponent doesn't matter if it requires something you don't have. For example:
A black mage with no MP won't kill anyone with the Death spell they lack the MP to cast. (Lack of means.)
The black mage won't kill anyone if noone ever gets in range of their Death spell when they're able to cast it. (As opposed to when they're out of MP/range, or "Silenced" or such.)(Lack of opportunity.)
If they don't know they can use their instakill, or know they can use it, but don't know HOW to use it, it won't do them much good. (Knowledge.)
& if they're unwilling to use it, perhaps because they refuse to kill, or they're more likely to use other tactics in-character-ly, then it also won't help them. (Lack of volition.) This is why SBA for a long time, has had "willing to kill" as a default assumption, even if it's been one OPs can always choose to exclude.
In short, AP/Striking Strength can provide win conditions.... & having a way to win at all is arguably the most important. But sometimes so can abilities like Longevity or "hax". LS in its own ways, can also provide win conditions.
Intelligence is a wide topic. Some characters, even those with profiles, know lots of things, but which don't help them in battle.
At its best, intelligence helps make win conditions more achievable, & sometimes, can bring its own win conditions (Social Influencing, Preparation, doing stuff with the environment.), but generally, Intelligence require you already having a win condition to hone them with.
A genius can't build stuff from the environment if they lack the LS to carry it or the Speed to get away or the abilitie sto defend themselves, etc.
"Skill" is nebulous, but often just a subset of intelligence. If you're faster with something, it's a subset of speed, & similar seems true for Skill in regards to other statistics.
Arguably, Stamina is the most important at extremes. If you have no Stamina, you can't take action (At least, not within a combat-feasible timeframe; Maybe you regain Stamina absurdly fast & you're fine.). A participant who can't do anything to advance their win condition may as well have already lost, but sometimes "passives" get around that.
So I'd say:
1. Character behaviour; What they'll actually do, regardless of their statistics, P&A, etc. dictates what they'll do. If a supergenius never uses their win in-character, then it only matters if they're "bloodlusted" in the match conditions, & even then, that may not solve everything. If they have options they won't use, those options may as well not exist, even if they can "use" them. "Passives" are much touted for a reason.
2. Certain Powers &/or Abilities, especially depending on who has what; Some, even individually can allow or disallow entire win conditions. Stuff like Non-Physical Interaction or Acausality Type 4 can radically influence win conditions.
3. Stamina. You won't do anything if you have no Stamina to act with. Thus, it's a win/lose condition all its own, & it matters a lot to using your own. However, the more of it you have, the less it seems to matter. But if you don't do what you need to do before you run out of Stamina, you'll lose in spite of those options. Unless your powers/abilities bail you out.
4. Speed, to some extent; Being able to act before your opponent can do anything, or just act fast enough to negate what they do can be win conditions all their own. But doing nothing fast is still doing nothing; You need a win condition, such as hax to make use of with your skewed action economies.
5. AP/Striking Strength (Not actually the same thing, Striking Strength is purely physical, it should be noted. Also, AP is just your capacity to harm, & so neither actually indicates capacity for Environmental Destruction; Island Level characters often can't destroy Islands with single attacks, for example.) because often, when hax isn't a win condition, these end up being that.
6. Durability. Arguably equal with AP/Striking Strength; Those are only so high up because they can provide win conditions, but Durability can deny win conditions; Which ends up mattering often comes down to Speed or ability to affect each other ("Hax"). Being able to survive your opponent punching you until they collapse of exhaustion, or just long enough to defeat them yourself can be a win condition all its own.
7. Lifting Strength/Skill/Intelligence/Range: None of these matter without something to exercise them on. LS only matters if you can outlift your opponent, be that to pin, suffocate, drown, bury, BFR, etc. your foe. Skill depends on what with, if what you're skilled in is relevant, & by extension, how that can be used. Intelligence is arguably the most important of these 3 because it helps use what you have better, but you need to have good options AND good intelligence for it to matter. If you're not smart enough to realize the win condition you have, being smarter than an animal didn't do you much good, did it? Range requires having a win condition to begin with, but at high enough levels, can make your opponent unable to interact with you, preventing them advancing any win conditions. But so can Intelligence, & in theory, maybe Skill, too.
Pardon all the words. Hopefully this helps. But yes, depends on a lot, & Speed is often equalized for a reason.
(Even if being faster means a lot less if you don't do what you need to win. It provides so many win conditions because most characters in matches have wins via hax or stats in spite of equalized speed.)