In NYC metered yellow taxis are not dispatched--you can only hail them on the street or at designated taxi stands (located at airports and a few places in Manhattan). If you call for a cab in NYC, you are going to get a livery car without a meter, not a yellow taxi, and the fare will indeed by a flat fare. It's pretty typical for the fare not to include tolls. Whether or not to tip is another question.

$60 all-in for a 22 mile trip with a tolled bridge and tip does not sound like overpaying to me. For comparison, it's about 20 miles from JFK to most parts of Manhattan, and the fixed yellow taxi fare for that route is $52 + tolls and tip.