It is an interesting idea, but it would have to be done in a way that makes sense and gives you an end product that is useful. I think having separate output transformers is the way to go. For the speaker cabinet, you could just use one and use something like a Weber Amp Switcher
https://www.tedweber.com/two-head-fs which would safely allow you to idle the unused amp section. You can only use one at a time however unless you bring two cabinets everywhere you go with the amp.
Years ago my first project was a Conn organ power amp chassis that had one power transformer and 3 output transformers. It had 3 separate sections of power tubes and phase inverters too, one for each output transformer. Likely they had a stereo pair of outputs running at higher power to run external Leslie speakers or something similar, and a mono internal amp of lesser power to run speakers inside the organ itself. Of course me being a glutton for punishment but also having a desire to learn and do something different, I made a separate preamp chassis with the intention to have 3 amps in one head. One was going to be a Blackface bassman clone with bias wiggle tremolo, another was going to be a JCM800 clone, and the third lower power one was going to be my own creation. Rev1 of the chassis had lots of issues, as did rev2. Rev3 should work well, but I lost my access to a chassis finger break so I can't make the chassis up. I also went back to school to get a BSEE in 2010, graduated in 2015. So I shelved it for the last 10 years or so. I do intend to get back to it though.
I've shared a few pictures with the power amp chassis along with rev1 of the preamp chassis, which was the configuration the last time I used it as intended. The Bassman channel sounded fantastic and worked well unless I had the trem oscillator tube enabled, then it messed up everything for the whole amp. It worked well up to about 8 on the volume and then started oscillating. My own creation worked fine also, and that one worked all the way up but it didn't sound that great.

The JCM800 channel didn't work at all and I'm not sure what I had wrong. I drew it all up on huge paper schematics along with a full size paper chassis layout for each of the sections before I built it. It uses 15 tubes, which I'm perversely proud of since it is one more than a vintage SVT, but it is also very heavy, and would require me to carry around one speaker cabinet that would work with all power levels, plus something like a Weber Amp Switcher like I mentioned above. I happen to have a Weber Fourhead which allows me to switch out 4 heads to one cabinet and have each unused section safely idling while not in use. It was intended to four separate heads I am sure, but it fit my need. I added some kits from London Power to keep the bias within reason when one or two sections are not in use. It scales what is going on depending on how many sections are active because as you put one or two sections on standby the load goes down on the power transformer and some voltages adjust up. The amp was originally cathode biased from Conn so it would do that automatically, but I made all the sections fixed bias so I had to add these kits. I was going to add some input switching for the JCM 800 section, and had it in the amp as you see, but I'll be scrapping that when I get back to it. It made it too complicated for little benefit and too much extra effort.
As a project it has been interesting and I have learned a lot. It's not really practical but that's fine. When the sections worked it sounded very nice and I will say the Foster transformers used in many old Conn organs are fantastic. Andy Marshall from THD first told me before I had started on the project that the big transformers on that chassis were Foster and said they were great. After I heard it I believe him. I'll finish it one of these days. I've done my level best to isolate the sections from each other and to use good grounding techniques. With the rev 3 preamp chassis my design issues from before should be sorted. I'd probably do it again, just because I like a challenge, but it is much easier to make a single amp in a chassis for sure.
Greg