If you walk in the street, it is normal activity on any day. Unless you are going somewhere very specifically, you would not use public transport. When you use public transport, you are going to work or shopping. It I very unlikely that you would just take a “stroll” on the trains!
You could be taking the train to shul, to visit a sick person, to visit your mother, or any number of muttar things which are too far to walk.
I have no reason to look it up, as I have no idea if he is reliable anyhow. I would perhaps be curious to see the haskamos.
In fairness, I think he was dealing with people who might use cars to come to shul- hence, much better to take public transport. However, the point is that there is no specific issur and one could rely on this, “beshaas hadechak”.
You are now singing a completely different tune, as all along you seemed to be saying that it is an optional chumrah to not use trains and elevators, and now you finally admit that it is only in case of great need.