The poop problem is certainly made easier by the fact that the overwhelming majority of New Yorkers poop in toilets, in specially designated rooms, that are directly connected to the sewers. And the poop gets flushed out of the toilet immediately. We also don't generally make humans work without bathroom breaks.
Decomposing poop takes time, and it's pretty awful to be around while it's happening. Also, for something to decompose into the ground, it needs to reach the ground, which it's not going to do if the street is paved.