The fear that I've heard described by politicians in my city is that uber drivers could position themselves in areas where they know from experience they will be most profitable (eg. dense downtown core) while avoiding areas that over time are not as profitable (eg. sparse outer areas). This fact, coupled with the fact that they are not obligated to travel to a certain area to receive a fare, means that certain areas could receive less reliable service as drivers chase high profit areas and are able to ignore others.
Your politicians are lying to you because they are paid off by the taxi monopoly. Service everywhere increases because of Uber and Lyft. The behavior you describe is exactly what taxis do. If you look at how things improved in all cities, service improved by orders of magnitude. In NYC, service to all 5 boroughs improved dramatically because of Uber and Lyft, because taxis would only service manhattan.
Before you follow blindly what your politicians are telling you, do some research and educate yourself first. Don’t rely on headlines and word of mouth.