I think you just didn't notice the teleporter menu screen. You can select your destination after you select the teleporter.
i don't have a teleporter menu.
it just says "egg corridor" ... "do you want to go the egg corridor? [y/n]". The computer next to the teleporter says "connected to grasstown" so this doesn't seem intentional.
I do believe you have to talk to Sue in her Cage before it will let you go. Otherwise how does the protagonist know why he's going to Grasstown?
talked to her like 10 times, running in and out of arthur's house but that didn't change it. Sue just says something like "you can use the teleporter in arthur's house to get to grass town, please save my brother" every time.
That would mean you'd have to either restart Cave Story from the begining or get a profile.dat save from the Tribute Site.
I'd just talk with everyone and such again, just to make sure.
Restarting Cave story wouldn't be that bad, since getting there doesn't take that long if you know what to do (this barrier in the egg corridor took quite some time...), just a little bit annoying.