As I recall, the tape reader switch had 3 positions - stop, auto, start. I think Stop and Auto were static positions, while start was momentary (spring loaded return to "auto"). In auto mode, the teletype would respond to Ctrl-S and Ctrl-Q in order to stop and start (respectively) the tape. (this may have been the origins of the XON/XOFF protocol used to do software flow control on serial ports - not sure which came first).
If Ctrl-Q (XON) does not start the tape with the lever positioned on "auto", and manually pushing the lever to "start" and releasing does not start the tape, it sounds like there is some basic failure of the tape read mechanism. But again, I'm getting out of my depth. Is there any sort of mechanical activity when you push to "start"? Or is it as if the mechanism is disconnected? You say that it works in local mode, which is odd because the mechanics are the same. I wondered if there was some hardware handshake going on with your unit, whereby the computer needs to raise CTS or some such thing in order to allow the tape to run. I'm not sure if anyone ever rigged up a TTY to do that, I never saw it but really did not work with many situations. Are you able to communicate from the keyboard to your computer, and computer to print/punch unit?