Just for reference, gitter.im did just that a while back. It's really nice to be able to join a gitter channel using a client of my choice and not a crappy web interface.
Also it appears Gitter is actually not operating a matrix server, but is instead bridged with https://www.matrix.org/docs/projects/bridge/native-gitter-br... ? Does that mean gitter.im is only reachable from matrix.org?
Personally when i had to join a gitter room, i would join from my XMPP account using the @matrix.org gateway (which despite being unreliable is very nice) so i have no clue.
No, bridged rooms generally are reachable from any instance. It's just that the bridge talks to matrix.org and matrix.org "hosts" (that's not 100% accurate afaik to how Matrix works, but close enough for this) the room just like any other matrix.org room, that is also reachable from other instances.