How much of balance is core strength? Which muscles?

[Query]

I'm still trying to recover my strength after a period of inactivity.

I notice it is much harder to balance (in bare feet) on one foot than it used to be. I clench my abdominal muscles to do it. To some extent I do that on skates too, though of course sideways balance on ice is much easier than on the floor, because the blade digs into the ice, and forwards/backwards balance is easier on ice - or at least on figure skates - because the toepick and tail tend to stop forwards/backwards tilts.

(Specifically, I clench my muscles to stabilize the joints in that area. In terms of physics, this effectively increases my moment of inertia, so it gives me more time to compensate for tilting. I suspect I am simultaneously clenching my lower back muscles at the same vertical level, but I don't really feel that.)

Should I be able to balance without much core strength? And be able to mostly use other muscle groups instead instead?

(Yes, I realize that if I completely relaxed my abdominal and lower back muscles, I would collapse onto the floor. But is it possible to just use the extent of muscle strength needed to maintain a standing posture, and use other muscles to create balance.)

As an example, I mostly balance a kayak (paddled in sitting position) by tightening those abdominal and presumably lower back muscles, and leaving my hips loose. So if the boat tips sideways, my upper body passively moves over the center of the boat, which usually is sufficient to keep me upright. (I sometimes use my paddle to stay upright too. But that wastes energy and makes it harder to turn and propel the boat. Weight balance is usually sufficient.)

But I've never been able to balance boards (surf boards, stand up paddle boards, wind surfers). Maybe that works better with a different set of muscles? I'm reminded of what someone told me, that tightrope walking and slack line walking uses and strengthens substantially different sets of muscles - though I admit, I don't try to do either.)

What group of muscles are best for maintaining balance while ice skating?