I just tried Square Times for the first time, and got it. Could be that I was very lucky, or that you’re missing some mechanic or trick.
Track looked like this at the moment of getting the achievement:
That was with two interchanges. Not sure if that’s normal on the NYC map, but at least one seems absolutely critical (the second one probably didn’t do much). The main funny business I did was teleport the green double carriage train from the square all the way to the eastern interchange if too much junk started piling up. (It was pretty much where half the map’s production had to go through). Probably had to do that around 10 times.
And obviously all those unconnected stations were occasionally part of the network. But the purple line was so obscenely long already that I alternated between two configurations for the eastern half of it rather than try to keep all the stations on the line all the time.