The induced demand argument works for trains too. If NYC didn't have MTA (no subway, no LIRR, no MNR) then the population of NYC would probably be 1% of what it currently is. Building more train tracks and having better train services also encourages more people to move to NYC so that these new train services become more utilized.

Neither roads or train tracks solve the traffic problem.

Train density is high enough that you might actually be able to build enough tracks to keep up with demand. Tokyo has just about kept up with growth by building trains, and (unlike cars in NYC) the trains don't have to dominate the city to do that.