I'd like to make a comparison with pole vaulting, specifically pole vaulting with a rigid pole.
(A search with search terms such as 'evolution of pole vaulting' gave images illustrating pole vaulting with a rigid pole.)
In pole vaulting most of the height gain is from conversion of horizontal speed to height. (Some of the height gain is provided by muscle strength of the arms.)
I get the impression that in high jump the jumping leg doesn't bend all that much. The leg muscles of the jumping leg do get to contribute some height, but not all that much, it seems.
I get the impression that to an extent the jumping leg is utilized as a pole.
I don't know how elastic human tendons are, but possibly some of the conversion goes via storing elastic energy in the knee tendons. (For comparison, kangaroos have a specialized achilles tendon that is elastic in just the right way. When jumping around the kangaroo's muscles do not need to contract, they only need to resist elongation, which costs much less energy than active contraction. The elasticity of the tendon allows the kangaroo to reuse energy, rather than expending energy at every jump.)
Much of the mechanics of the high jump is in the swing of the free leg. It seems to me: the more vigorous you can swing up that leg during takeoff the more height you can gain. Again this is conversion of horizontal speed to vertical speed. With the running approach the free leg has already some horizontal speed to begin with, thus providing more available initial horizontal velocity to be converted to height.
Still, the jumper cannot afford to convert all horizontal speed to height. Since the jumper clears the crossbar with his back turned to the bar his path over the bar is necessarily a very diagonal path. The vertical acceleration of the jumper's center of mass is a given: that's gravity. So for the highly diagonal path more horizontal velocity is needed than when clearing the crossbar at right angles.
Conversely, when jumping from a standstill position some of the precious energy has to be expended on causing the horizontal velocity needed to clear the bar. Not as much horizontal velocity as with the diagonal path, but still.