As far as the cap applying to all boroughs; that is because no one would sign up for a license that only went to the outer boroughs. The yellow/green cab experiment failed; introducing a similar dichotomy to black cabs would probably have the same results.
The yellow/green cab experiment failed because it happened before the magic of smartphone ubiquity. The problem with yellow/green cabs was density -- the density outside Manhattan is thin and the area is enormous. The chances of a yellow/green taxi just passing by when you need one is close to zero.
Uber/Lyft solved this issue with beaconing. Uber/Lyft further reduce cost if you pool, which is great all around.
Poor people in New York overwhelmingly don't drive. They don't even own cars. The median car commuter into Manhattan makes far more than the median transit user. https://www.ibo.nyc.ny.us/newsfax/insidethebudget154.pdf
As far as the cap applying to all boroughs; that is because no one would sign up for a license that only went to the outer boroughs. The yellow/green cab experiment failed; introducing a similar dichotomy to black cabs would probably have the same results.