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.
As far as I know, there's no reasonable way to bike between middle/lower Manhattan and New Jersey. A new tunnel / bridge should definitely add some bike lanes!
This is known as The Gateway Project[1], has been in progress for many years and with many years and billions left to go, and was listed in Trump's "top 50" infrastructure plan.
It's a long distance. The GW Bridge was seen as a "world wonder" level engineering achievement when it was constructed.
1: https://en.wikipedia.org/wiki/Gateway_Program_(Northeast_Cor...
The 401 and QEW are a disaster though, I agree. Mississauga and the rest of the Western GTA sprawl are the worst-case example of how induced demand can ruin a roadway.
Hopefully some events tip the political scales, as the open hostility of the Federal government will cause irreparable harm.
Maintaining things like school districts, towns, villages and special districts will always be local priorities that always win against regional initiatives. You probably have 50 police departments alone just on Long Island, with at least 50 collective bargaining agreements that would need to be consolidated and dealt with. That's 50 police chiefs who need jobs, and thousands of generously paid policemen who would need to integrate with a larger and less generously paid NYPD. You'd have the same situation with school districts.
The formation of "Greater New York" (New York City) was only possible in the 1890s because the political machinery was in place to allow annexation and takeover to be effective. Tammany Hall was more powerful than the opposition. State administrations in New York have been trying to encourage consolidation and shared services for decades with very limited success. Even dying places in Upstate New York want to hold on to local autonomy.