Funny that you mention that, becuase I had an idea last night combining the desire for more traditional beasts of burden, and technological assimilation.
Instead of assimilating technology from capturing settlements, assimilate parts of the other tribe's culture from interacting with them over time. The difference might seem subtle, but a key difference is that you could assimilate parts of a tribe's culture that you're completely friendly with. Trade and negotiate with another tribe for long enough, and some of their culture will start to bleed over.
Part of this could be straightforward technology, but it could also be more subtle. For example, one of the races I want to add has a strict noble caste, and some buildings can only be operated by a noble. Humans interacting with them over time could have the opportunity to form a noble cast of their own.
If you take that, and mix it with the Horse Nomad race that I'd like to add you get an interesting possibility. Humans at first only have Pigs for beasts of burden, but if they interact with the Horse Nomads they could assimilate the ability to raise horses. Horses would give a direct upgrade for trade wagons and chariots. This would effectively be advancing your own culture through diplomacy, which I don't think I've ever seen before.
No promises that this will happen, but I find the idea really compelling.
Edit: for the misbehaving wagons, could be the fault of the pathing optimizer, or it could be that they're actually following the shortest route. Roads only cost 75% of the movement speed of open ground. If they had to travel twice as far to use the roads, then that wouldn't be worth it.