Only the top 4 (or however many they tested that year) worked off without stirrups. But generally the top contenders are able to jump without their stirrups, or should be able to, since it’s one of the possible tests.
No stirrups is just an exercise, not a fix in-and-of-itself. It’s suggested often because when it works, it works (and it’s never a bad thing to be able to ride without them in case you accidentally drop them). And when it doesn’t, it re-inforces bad habits because you compensate in the wrong ways for lack of core/inner thigh strength/balance (or you over-do it and pull a groin muscle – don’t do that either!). When you’re cantering (or walking) without stirrups, it’s good to pay attention to what your leg is doing too.
With my trainer, for her older or returning riders, she had them work towards 1 good lap around the ring, at the posting trot or 2-point, in both directions. But it’s not expected that anyone gets it right away. And 2-point with stirrups is another good exercise to get weight down to your heel. IMO, it’s not so much that you need your heels down as far as they will go, but because that weight lengthens your leg, which allows you to wrap it around the horse’s sides. When a rider grips with their knee or lower leg, the leg often rides up to propel rider upward out of the tack, instead of staying down and around the horse.
Try this: at the halt (which is a little counter to what we were discussing, but hear me out), sit on the horse as you would normally. Then arch your lower back/stick your bottom out behind you – that’s all you need to do to post or get into a half-seat. It’s not a big “up” movement with your lower leg or knee, it’s mainly the thighs that are doing the work. At the trot, when the horse’s inside hind leg steps under, think close your hip angle (with your hips/bottom “out/behind you”, not your shoulders forward), instead of lifting up.
This is again kind of ridiculous-looking/feeling, but if you have a full length mirror at home, stand in front of it, so your side is in view. Get into riding position, align your shoulders, hips and heel, by bending your knees. Then close your hip angle by sticking out your bottom (don’t move your shoulders) – that’s the “post” or half-seat position. That’s all you really have to move. The horse’s movement does the rest, on the flat and over a jump (and your inner thigh-to-lower-leg determines your balance on the horse and ability to move with him).