The single tunnel shared by Amtrak and NJ Transit trains is 100% full of trains which are over 100% full at peak.
So yes, PATH and other trains are extremely useful.
Whereas Manhattan subways have two tracks per tube, the PATH has only one. Service is not only slower, but you usually wait significantly longer just sitting on the PATH before it even leaves the station.
Hurricane Sandy also did irreparable damage to the tunnels crossing the Hudson.
I believe you are correct about the tracks through the cross river tubes, but less traffic makes it easier to incorporate in NYCTA, not harder.
They talked about running PATH trains up the rest of the 6th Ave trunk in the early Aughts, but it fell apart due to political dysfunction (shocking).
The cross-harbor tunnels are cast iron, and are not straight.
The PATH and NYCT systems are incompatible. I think they use the same track gauges, but NYCT cars (either division) are too long to navigate some of the curves on PATH.
The effects political dysfunction are really starting to materialize now. Penn Station, North River Tunnels, PATH tunnels are all in really critical states now.