I have moved this thread to the succulent forum, where it may reach more eyes.
It is surprising to me that someone would want to cut up a 15 foot tall ponytail palm, given baby plants are pretty common in nurseries. But hey, when you want more plants, there's usually more than one way to go about getting them.
Flowering is often a trigger for branching, or at least they are associated on mature plants. So it's not surprising you aren't seeing branches if you haven't seen flowers. That could be related to water, maybe?
This plant should be relatively easy to start from cuttings if they aren't really tiny. I don't think rooting hormone is required. In AZ you have to deal with some pretty brutal summer heat, so you need to time this sort of thing for the more permissive seasons. Be aware that plants grown from cuttings will have a different form than plants grown from seed.
More about the plant, including details relating to shape and fertility, here:
https://garden.org/ideas/view/...
This species has male and female plants, so even if your tree does flower, it would have to have a close encounter (via pollinator) with a plant of the opposite sex in flower at the same time, in order for one of them (only) to produce seed.