The only exception to the following are cities where car services not arranged by the hotel are dangerous.

In all other situations, you should feel free to book with whomever you please. Many hotels book with the same car services as you might call and then add on multiples of the fare. It's not uncommon in NYC, to have hotels charge $250 for a run to LGA which costs $55 in the same car when booked individually.

As to Uber, it's my service of choice and I would under no circumstances worry about summoning one to the front door. The bellman gets the same tip, from me. Perhaps not from the car service. But, that's not my issue.