Sorry, but not true. One mix, routed to two different 4-channel signal paths at the same time: a 4 track 1/2" tape machine for the discrete master, and the other one to an SQ encoder that was then routed to a stereo recorder that produced the tape used to make the SQ LPs, etc. CBS felt that the less tape generations something went through the less chance of inducing phase errors that could compromise SQ decoding.
One of the (many) weaknesses of SQ was that it couldn't deal with mixes that had excessive (or even much) reverb or echo in them, which is why pretty much every quad mix that CBS did at their own studios in New York and San Francisco is dry as a bone compared to its stereo counterpart. It's also the reason that Wendy Carlos didn't do any more quad after
Switched-On Bach - when she heard playback of the decoded SQ version of the album, things from the discrete master were simply eliminated in the process of decoding the album and she felt that was unacceptable (she complains about this at length on her
overly-verbose website).
The quad mix for the
Burton Cummings album was done at Studio 55, a studio that (as far as I know) produced no other quad mixes, so the guys who did it presumably had little (or maybe even zero) experience with quad mixing, much less the vagaries of the pitfalls of the SQ system. CBS NY had a setup where they could flip a button and hear what their mix would sound like through an SQ encode/decode process, but who knows if Studio 55 had anything like that - somehow I doubt it. I think they did the mix how they wanted to do it for the discrete master, and didn't really think that it might affect the SQ decode, which is exactly what happened - the SQ decode "eats" some of the dry signal in the mix (ie the lead vocals, instruments, etc.) and what's left behind is signal that's more reverb than source.
Quad mixes were done under time constraints and tight budgets - it was already a big enough step for labels to be doing a second mix after the stereo mix, they weren't paying studios to do the quad mix (which sold, if they were lucky, 10-20% of what the stereo version sold) twice, especially in the dying days of the format in 1976.