The government and taxpayer could then take any profit (or operate at a loss / break even if they want to subsidise costs for their citizens). The customer acquisition / marketing costs would be almost zero if the government mandates it as the only or preferred ride-hailing app.
I don't really think Uber is going to innovate much from here (e.g .how much has the uber experience changed since UberX was rolled out?) outside of self-driving cars etc. there's no reason why we need a private company to innovate further, and there's almost no risk in doing this.
It would be seen as a natural extension to mass transit, and every government around the world should be providing it for their citizens.