Yeah they use a substantial amount of proprietary middleware now, so even if they wanted to open source the engine it would be difficult to pull off. As of Doom Eternal they're using middleware for at least audio (Wwise), video (Bink), compression (Oodle), physics (Havok), culling (Umbra) and networking (Playfab), plus any references to console SDKs would have to be meticulously scrubbed from a public release, which was less of an issue back when they were a PC-first studio.
Rage should still be releaseable as Carmack wanted to release the source code at some point but he left id before that happened.
I always found amusing that to replace him Zenimax built a whole new studio in Germany to hire the top programmers from Crytek and even then Doom 2016 used a litany of middleware :-P.
Shame they never finished Rage. It was a pretty good game up to the point where the boss came in and said they needed to have it out the door in a week.
It probably went through some heavy development hell, it was id's longest game in development (7 years) and IMO it wasn't really that great as a game - the shooting bits were great and felt very good (the best part of the game), the driving sections felt bad with the levels being too small to drive around and too large to walk and the quests felt tacked on. The various "bonuses" (like those things that hang in the air and you can pick up by making stunts) feel like excuses to pad out the game's length.
It was like the game didn't know if it wanted to be an FPS or some seamless open world ARPG so it tried to do a bit of both and the only reason the FPS side didn't felt bad was because id was masters at making FPSs by that time - but it was certainly dragged down.
It would have been way better if it was a regular FPS with a linear progression of levels like id's previous games.
Doom 3 even had that issue to a smaller degree. They had to rewrite some sort of lighting middleware before open sourcing. That's just one piece of middleware though, and modern stuff is probably much more hairy and complex.
That one wasn't middleware but rather a patent issue, the code in question was written by Carmack but he had unwittingly rediscovered a technique that had already been patented by Creative Labs. Id licensed it from Creative for Doom 3s commercial release but legal wouldn't sign off on including it with the open source version so Carmack swapped it out for a different, slower algorithm.
It could just be that they started using other proprietary components in their code though.
Would've loved to read the source code for Doom 2016!