Boots are favored over polos for jumping. Both only provide protection, not support. Boots give more protection than polos and can't come undone as easy. You also can't use polos if you are riding anywhere wet. Polos are harder to put on correctly. If you dont know how to wrap a leg, don't use polos.
You dont need anything in their legs to jump, especially lower heights. I generally don't jump mine in boots. I've used leg protection more when teaching laterals or changes than over fences.
They can provide protection against an overstep on landing or a knocked pole from hitting the leg. When the horse is landing and their front fetlocks are fully extended, a small amount of trauma to the back of the tendon can cause significant damage. Those kinds of accidents aren't terribly common tho.
Boots can hold heat on the legs, but for the span of a normal schooling session it shouldn't be a concern if the boots come off right afterwards.