My Model III doesn't have a power supply problem.
You said the screen image is getting wobbly and it's not starting up reliably. That's very likely a power supply problem. (Possibly compounded by aging capacitors in on the monitors' analog board... but hey, I forgot, you have no interest in actually fixing any of these problems.)
Regardless... "Concoct and defame"? "remote and exogenous desire"?
Anyway, if you want something directly relevant beyond what a couple people who know what they're talking about have already contributed I'll attempt to clarify it a bit: Yes, the pinout on the back of the 12" CRT tube itself in a TRS-80 is *probably* the same as the 9" in the Compact Mac, but given what appears to be your tendency to conclude that if the plug fits it should "just work" that's frankly probably a dangerous thing to tell you. After all, you said this:
The issue would be whether that uniformity belies differences in voltage or polarity - or some other standard [i'm honestly curious how the graphics board will control for size and resolution, since the onboard (/30 analog board) controls are only vert/horiz but not size].
Do you think picture tubes are somehow "plug and play" and the board will just magically figure out what you've done and adjust everything automagically? Because, no, that's not how it works.
Anyway, if you were to thread the deflection yoke attached to the Mac's analog board onto a 12" CRT, which it may or may not fit depending on the model of each, and power it up you *may* get a dim and out-of-focus picture; as Technight noted the flyback transformer on the Mac's board quite likely won't completely make the larger CRT happy and/or might burn out trying. Further, the Mac's analog board itself is mostly "hard-wired" out of cheapness, IE, it's short on adjustment pots, so even if the analog board does have enough reserve in it to power the larger tube adequately you'll probably have to change out some resistors and other components to get a clear, centered, and properly sized picture. (And obviously the resulting picture will have the same 512x342 resolution that the 9" CRT shows because *that* is dictated by the digital circuitry on the motherboard, which likewise will have no idea whatsoever that you changed something.)
(As for using the "analog board" in the TRS-80 to drive the monitor by connecting it directly to a Mac motherboard that's also pretty much a non-starter for a novice. The TRS-80's monitor runs at NTSC frequencies, IE, about 15khz hsync, verses the 22khz the Mac runs at. It would need substantial work to sync at the faster rate and further the output from a Compact Mac motherboard isn't "normal" TTL video, it's a little "odd"; there was a thread about that a few months ago which boils down to even if you have a Multisync monitor that can sync at the Mac's frequencies you need a small amount of additional circuitry to turn the "raw" drive signals from the Mac into standard h/vsync signals.)
In short, this is an awfully ambitious project for someone who's afraid of soldering irons and only learned recently that the part of a CRT with the wires coming out of it wasn't something you could buy an ala carte replacement for when you let air into it.