If you're trip was solely along Rt. 119, are TZX drivers allowed to let you on at the Tarrytown rail station?
Would TZX operators let you get on to go where? East on 119? I don't think they operate on 119 anymore, they take I-287 east to Exit 5 and then go into White Plains. During the strike a few years back Rockland buses would not carry anyone strictly within Westchester. Is that still the case? I don't know.
As for Elmsford, the buses all connect at the intersection of Rts. 9A and 119. There is no transit center there, just bus stops on all four corners. It's called Elmsford Square in Bee-Line timetables.
It's a commercial district and there is no available land to build a transit center on. Buses go in all four directions so even if land were available it would be hard to design a center that all routes could access without looping off-route (which riders hate!).
Second, I don't see too many people making transfers there. The one I see most is between the 5 and the rush-hour 27. People who live in Yonkers and work along Rt. 9A in Elmsford or Hawthorne. Next would be people like me, I guess, who go to/from Tarrytown on the 13 to the 1C/14/27 to jobs along Rt. 9A in Elmsford or Hawthorne.
But usually I only see five or six people at a time making the transfer.