The argument of regulators is that, in the absence of proper congestion pricing for cars on the road, the market equilibrium for number of taxis per inhabitant will be higher than is efficient. That's how they justify a cap on the number of cabs. (It increases wait times but, they argue, it is worth it for the fewer number of cars on the road causing congestion).
So the question isn't "are there more Manhattan pick-ups after Uber entered the market?", it's "are there more cars on Manhattan street?" or "Have average trip times in Manhattan increased?".