https://en.wikipedia.org/wiki/Ljubljana_Marshes_Wheel
Wheel & axle made of wood, for a two wheel push cart.
Spend some time using an ordinary dolly to move heavy boxes around, then try carrying them. There's just no comparison.
Besides, it's one thing to say that wheel technology is gated by metals technology, quite another to say it is gated by the existence of pack animals.
I’m pointing out projecting modern needs backwards out of speculation isn’t likely to yield much understanding of why someone didn’t invent a handcart in precolumbian mesoamerica. Jared Diamond at least compares the material conditions of the two sets of continents—that seems like a decent tactic for approaching the past, even if I do have large complaints with his confident narratives.
I've been to Chichen-Inza. It's flat.
The Egyptian chariots had wooden wheels and axles, and were light.
https://www.brown.edu/Departments/Joukowsky_Institute/course...
In any case, the baggage cart like you're describing, or maybe a hand truck, seem to be both artifacts coming out of industrialized society. I'm not sure how useful one would be on a steep hill if the cart itself, including the axle and wheels, were made of wood. You don't really see modern, usable baggage carts until the industrialized production of steel (I'd love to be corrected). Hand carts? Sure, but of a more significant weight:load ratio meant for longer distance, heavy load, flat hauling—remember these suckers often have brakes you wouldn't want to engage on a runaway cart, like a peg through the spoke or axle or a wedge under the wheels.
I'd think some kind of carrying + strap frame like you see today would be a better fit for personal hauling and balsa might even work well for it.