I am not happy here with my own answer, and I am neither an instructor nor even a student of functional musculo-skeletal physiology, but possibly some thoughts might help a bit.

Firstly, the patella is by now pretty confidently regarded as a sesamoid bone, originating in the tendon as a reinforcement, and developing from hyaline cartilage largely in reaction to stresses on the tissues. It is fairly clear therefore, that there are stresses in the tendons and ligaments of the knee that do not match those of the elbow. (The fact that by now the patella is such a major item in our anatomy that it is already prominent in the normal foetus, does not nullify this observation.)

Secondly, although both knee and elbow joints happen to be where two bones meet one, there are differences. Firstly in origin. As far as I can tell all tetrapod vertebrates have legs based on the same origins and what you might call bauplan, or basic architecture. This is partly because of the anatomy of our first fish-like amphibian ancestors, and partly because of the way that our later ancestors' limbs developed from the ancestral form.

This gave rise to a complement of four limbs (not counting tails), the front two oriented forwards with largely a polling function, and the hind two rearward, largely for pushing. This has landed you, me, tortoises, tigers, toucans, tamanduas and Tachyglossus with what we know as elbows and knees. I should not be at all surprised if there are proprioceptive and other control functions to the patella and its connections, but I find it hard to imagine that such functions were the basis to the development of the patella, rather than adaptations along the way.

I cannot offhand think of any tetrapod that developed stronger forelimbs than hind limbs before it had already developed knees and elbows. The hindlimbs accordingly were stronger and had developed more pulley-like knee tendons. Working around the corner as they did, those tendons were prime candidates for developing sesamoid bones. The entire structure not only was strong, but permitted good control of its position without much question of slipping out of position, which is a perennial problem with pulleys. It also gave an improved angle of application of forces to the whole structure.

No doubt a similar structure in the elbow would have worked as well, possibly better for all I know. But by the time the question might have arisen, we already had elbow joints, or at least our ancestors did. It seems to me as likely that we did not develop an elbow sesamoid because we had by that time already established an olecranon, or whether the olecranon developed because of a slowly increasing tendency to apply tensile forces to the head of the ulna. In either case we wound up with a comparatively short, light, straight tendon between the triceps and the olecranon. This is not a favourable situation for developing a sesamoid.

Bottom line? We (or our ancestors) began with two different joints, with two different kinds of function. These resulted in different shapes and different forces. One favoured the development of a sesamoid bone, the other did better without. It would have been perfectly conceivable for either kind of joint to develop in either place, but once the basic architecture was established, it would have taken a major leap or at the least a very roundabout adaptive route to change the situation.

Handwaving?

Certainly, and I am sorry for it. With any luck someone else will come up with something more persuasive, if not actually more cogent.

Let us know if you would like to take this discussion any further.

Thanks, go well,

Jon