Uber has the advantage of being ubiquitous among a certain class of travellers - this obviously is not an insurmountable advantage, but it does mean I naturally tend towards using Uber when travelling unless there's a particularly compelling local alternative.
Everything else about your taxi service sounds good, and I'd be more than happy to use them, I'm just lazy. (And from other comments, I see that there's taxi company apps like FreeNow which seem to be gaining regional market share?)