Many modern chips do in fact use signed firmware yes.

Technically they could publish the source in that case but I think some patents would prevent them from doing this. Radio stuff is riddled with patents. Also most likely the "why would we" reason. There is no benefit for the manufacturer.

Why would patents prevent open source? As in, they're afraid of patent trolls trawling their code for "infringements"?

If it's concerning their own patents, no need to hide the source code. A patent is literally "letters patent" or public description of an invention. Trade secrets on the other hand I could understand.

It's often a condition of licensing those patents. NDA's, they're huge in the radio and embedded world. It only became a bit more open when the Arduino and Raspberry projects came along. And even there it wasn't fully open, especially with the Raspberry because Broadcom is one of the worst offenders.

I'm surprised how much they were able to open source as-is. I think part of that is that the SoCs powering the raspberry are kinda 'old news', definitely not the bleeding edge kind of embedded tech. The first raspberry basically happened because Broadcom had a whole bunch of old chips they weren't able to sell. Only when it took off they started making some actually for the purpose.