IIRC there was mention of it providing some p2p network style communication capabilities for Ukraine's military, and one of the reasons it's appealing to the US's military is the ability to route communications entirely within the network (well, with the gen 2 satellites which have laser interconnects).

So it can (at least eventually) function without 'regular internet', although I would still be hesitant to call it a viable infrastructure choice if the goal is to get around government control, simply from how much SpaceX have to appease the government to do anything space related.