I mean no matter what I imagine people will try and get the best breedable mount they can.
For an answer though, in real life why do people still have donkeys and mules if everyone could just have a horse? They require different levels of maintenance and handling, they are capable of different feats, they aren't necessarily suited to the same biomes.
Same is likely to be said for trison (or ursaphants) and horses. Horses will likely eat less, definitely require less space, perhaps they are easier to train as well. Maybe the taiga is too cold for horses, but trison can survive and then the janoan jungles are capable of allowing horses to guide through but trison cannot navigate due to space. Maybe you have to get information quickly to a neighboring county and trison aren't quite built for the speed horses are.
Again though people will probably go with what is best as long as they can afford it though, so maybe the answer to your question is some time in the future we see trisons everywhere and no longer see horses for common beast of burden tasks.