This is pretty much the TL I'm working on, durnit!
(But mine will be a bit different, and such

Not mad)
I think you underestimate the willingness of NASA to push for the Shuttle. Everyone at NASA, top to bottom, thought the Shuttle was the way to go--they were as much behind lower launch costs then as NewSpace people are now. The Shuttle was very, very near death several times, but each time managed to pull through thanks to the compromises NASA was willing to make--cut down on size? Sure, they would do that. (Although it ended up being unnecessary--but they would have built something the size of the HL-42, if they had to) Make it only partially reusable? Yep, on the table. Delta wing for AF requirements? Why not? I'm handwaving it in my TL, so I'm not too averse to you handwaving it here, though.
And you overestimate the willingness of Nixon--or any US president--to cancel HSF, effectively. If he's not funding new programs, that means US HSF is dead by the mid-70s, at the latest (by that point all existing Saturn IBs, Saturn Vs, and CSMs will have been used, and the production lines had been shut down by the Johnson administration in '68), and that simply isn't going to happen. Not in the Cold War. I mean, even now Obama is trying for commercial crew to follow up the Shuttle for US HSF capabilities. They're going to pay for
something, probably a follow-up Apollo. Big Gemini was a contractor proposal, so I don't think it's likely to be accepted, and so a modified Block III Apollo is much more probable.
There are some things with the astronauts that I think are off, but I'm not an expert on the Astronaut Corps (to put it more bluntly: I know less than the first thing about their personal relationships), so I'm not going to bother you about those.