But, I agree on the part that they will slow down a bit existing public transportation, but, if Uber served routes that are currently difficult to reach, it has public service as well.
Why would someone pay $10 for the Uber service, meanwhile the local one is just $3? There is a good chance that the local bus doesn't cover certain areas properly, or stops too frequently, making it a slow trip for regular commuters.
Ps. In Europe there is both public and private trains, both running the same tracks. I don't see a problem with this.