If you use a motor to cock the trebuchet, you have electrical potential->magnetic flux (don't ask me for the details here, I don't know the terms) -> gravitational potential. If you cock it by hand, you are consuming chemical energy to produce gravitational potential energy.

Regarding wheels, sure they effect the energy transfer involved, but they are no more of a form of energy transfer than addling more weight, a longer sling or any other tuning changes.

Personally, I think what happens to the energy after the throw is the most interesting and educational. We all like to teach conservation of energy, but eventually the projectile, and the trebuchet are both at rest, and with less energy than they started with. Where did it go (heat mostly), and why can't you get it back? Thats a real applicable and useful concept! Aside from being a fantastic introduction to entropy, information theory, and thermodynamics, it provides an understanding of efficiency in general, and is widely applicable. The topic of thermal losses in systems is perhaps the most generally applicable transfer of energy in engineering and physics, not a cop-out. In-fact, there is evidence it will be the limiting factor of the universe itself!

Also regarding the frictional and other "small" losses: Most trebuchets lose much more than half of the energy this way, so it is in-fact larger than the transfer to the projectile. And since they are basically the same as the transfer the projectile makes upon impact, (both are thermal) all the energy goes through this transfer eventually. Most of the sound ends up as heat too by the way.

One of my favorite application of this: Where does the energy go when you walk down a hill? (heat!) Since going up hill gets you hotter than down, this proves (after some minimal logic+math) that walking up hill is less than 50% efficient. Neat right?

Its pretty interesting (and quite practical) to compute how much kinetic energy was in the projectile and thus compute the losses/efficiency of the trebuchet. It might be a bit heavy on the algebra, but all the hard parts can be looked up instead of derived. My dad managed to teach me some of the basics of integral calculus doing that exact problem around that age (We derived most of the details ourselves), but I suspect thats not for everyone.