The key to heavy horse drawn carts was the horse collar. A horse collar fits over the shoulders of a horse allowing it to push a load with its back legs rather than pull a load with essentially just its front legs.
Before the horse collar (invented about the 5th century CE) horses were harnessed either with a yoke or breast strap. Because of how these fit on a horse they could only use about half their available power at best and breast straps could impact their windpipe. For heavy loads oxen were used since you'd need twice as many horses for the same load.
Horses were far superior to oxen for many types of work once the horse collar was widely available because they're faster than oxen and have better endurance. With a horse collar they could pull the same load as an equivalent number of oxen but faster and longer durations. It not only made heavy horse drawn vehicles practical but increased the cost effectiveness of those vehicles since they could have longer duty cycles.
Animal drawn vehicles involve much more than just wheels. Their use is very much about the technology available and suitability for the environment.
History shows us the wheelbarrow is not obvious (things which are obvious get done quickly), the chariot provides an intermediate step and thus makes crossing the gap far easier.
So the thousands of years gap between heavy carts and chariots vs wheelbarrows, isn't something you can explain by citing insufficient technology or unsuitable environmental conditions. I think it all comes down to nobody had that idea yet.