So I'm curious - the Minutemen haven't offered me any new missions since I went to them to get into the Institute proper. Desdemona of the Railroad keeps pestering me about siding fully with the Railroad or Minutemen...but all I have are radiant quests from Preston.

The Radio Freedom Tower at the Castle is bugged out - the operator keeps insisting the power isn't hooked up when it is. Has that borked the Minutemen completely for me, or do I have to turn down Desdemona's offer of full cooperation with the Railroad to advance along the Minutemen story?

It doesn't matter if you tell Des you're working with her- just don't do the quest she gives you after you tell her yes. You're all caught up with the minutemen if you've done old guns and gave sturges the tape. Tp progress with the minutemen you have to do the synth retention quest for the institute. After that you'll talk to father, and get told about the battle of bunker hill- at which point you can warn Des about the incoming assault. Fight the battle, then return to the institute and turn them mostile by refusing to work for them. Talk to Des again and she'll tell you the railroad can't take on the institute without you working on the inside, and that you'll need the minutemen to take them down. Finally, at that point, you can talk to preston and move things forward.
(Also if you want to take out the BoS you need to turn them hostile before you talk to Preston- kill a named member to turn them- like those blokes in the police station)