Mono and stereo compatibility is why it was approved for testing live on U.S. stations fifty years ago. End result was: Nobody cared.
You still haven't answered the question: Why would anyone do that? Quad was four demodulated discrete channels. 5.1 is five. In traditional analog multiplex audio, where would you put the fifth channel? There would also have to be a unique pilot to tell the receiver to decode surround. Oh, and since there are no generators or receivers to receive something like this, what manufacturer would build it?
Quad was rarely discrete channels, though. The matrices were full of adjacent channel crosstalk. 5.1 is a standardized speaker plan, discrete channels, and the center solves lots of problems.
There are no "generators" because we can't transmit the required bitstream bandwidth, and so no system has been standardized for audio-only broadcast. TV does it all the time, and there are billions of 5.1+ capable receivers in the world since the mid 1990s.
The two big technical problems with Quad was the invasion of two more speakers into the livingroom, critically placed, and the very tiniest listening window where it all worked, dead center amid all 4. And the matrix systems had unimpressive adjacent channel separation. Add that to the market confusion of which decoder to buy, which medium to buy (there were 4 systems on vinyl, 4 channel open reel tape and 4 channel 8 track tape) and a rather inconvenient speaker/listener plan, and it just never went far.
5.1 solves several technical issues, as we no longer use big full-range speakers thanks to the satellite/sub and bass-management, and with the LCR front arrangement, the phantom center issue of stereo is gone, as is the dead-center Quad problem. The Ls and Rs channels are now well separated, even in ProLogic matrix decoding. The technical issues that remained in the late 1990s push for multi-channel music related to the confusion of how to use the surrounds from a mix perspective (in the band or hall ambience), and if they should be direct radiators or diffuse radiators. But that could have been solved with metadata.
All of THAT stuff rolled into a whole new form of market confusion, though slightly less than with Quad (4 coding systems, with media compatibility issues). 5.1 has 3 primary coding systems, but they are invisible to the user, with decoding auto switched. 5.1 is still invasive in the living room, but speakers are smaller, placement is not nearly as critical as Quad was, and its highly standardized for film and video (standardized in ITU documents as early as the late 1990s).
When HD radio was being standardized, radio had an opportunity, as did the music industry, that just flew right by. The music industry was well into a downward spiral from file sharing. Two channel digital radio, at first impression, is not a day/night improvement over FM stereo. Multi-channel radio could have pushed the music industry into a big re-mix, re-release cycle that it needed, and multi-channel radio, in cars and in existing 5.1 systems in homes, could have offered that day/night improvement associated with justifying additional cost. But it didn't happen. 5.1 easily downmixes into two-channel stereo and mono, as part of either the Dolby Digital bitstream and metadata or the DTS bitstream, so we wouldn't have been making something incompatible.
But I believe what really killed the 5.1 music push was the perception that such a large market segment was on headphones that why bother? We didn't have a good cheap way to virtualize surround in headphones then, so it just flopped. Got to say though, that some of the 5.1 releases that did make it out were spectacular. Dark Side of the Moon is amazing, for example. Even Thick As A Brick is a revelation. There just wasn't enough of it.
Stand by for the
Beatles in Spatial Audio and Atmos, though!
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