As far as I can tell, it's the latter option. Rumble has a whole section of their website explaining how to use OBS with it [0]. The complaints seem to be entirely based around Rumble not being available in a drop-down menu [1]. It's kind of pathetic honestly.
[0] https://studio.rumble.com/help/guides/obs
[1] https://x.com/dhh/status/2108603279345070205
(Sorry for the X link, but that is the only place that I can actually find people complaining about this).