How do I download Mac OS Monterey installer for an older Mac with incompatible current OS?

The Mac / OS won't let you download a version of macOS that is too old to run on your computer. At least, not through the normal means.

See if you can retrieve it through Terminal:

softwareupdate --fetch-full-installer --full-installer-version 12.7.6

You would need access to a Mac that is compatible with Monterey.

macOS Monterey is compatible with these computers - Apple Support

What is wrong with the older Mac that requires reinstalling macOS? Most issues are caused by either third party software that can be bypassed by booting into Safe Mode, or issues are caused by hardware issues....perhaps the internal boot drive is failing.

If you provide the exact model Mac, then we can provide options. One option would be to try to boot into Internet Recovery Mode using Command + Option + R to attempt to access the most recent version of online installer that is compatible with the "broken" Mac. Or we can at least tell you which other Macs you could use to create a bootable macOS USB installer that would be compatible with the broken Mac. Since you are trying to use Monterey, that would seem to imply you would need access to another Mac from about Late-2009 to mid-2022 (perhaps even one from 2007-2015).

You can get the exact model of the "broken" Mac by entering the system serial number on the check coverage page (please don't post the serial number on the forum since it is considered personal information):

Check Your Service and Support Coverage - Apple Support

It is unfortunate that Apple makes this process way more difficult than it needs to be.

Drat! Thought the user may be able to bypass the App Store with that one.

If I run softwareupdate --list-full-installers, it only returns matches that will run on my M4 Pro mini.

* Title: macOS Sequoia, Version: 15.3, Size: 14889290KiB, Build: 24D60, Deferred: NO

* Title: macOS Sequoia, Version: 15.2, Size: 14921025KiB, Build: 24C101, Deferred: NO

* Title: macOS Sequoia, Version: 15.1.1, Size: 12428948KiB, Build: 24B2091, Deferred: NO

* Title: macOS Sequoia, Version: 15.1, Size: 12427940KiB, Build: 24B2083, Deferred: NO

Any other full version installer I try to download simply returns, Install failed with error: Update not found

If you run this command in the Terminal…

softwareupdate --list-full-installers

It lists all the versions available for download for a particular Mac.

That won't work…

"Install failed with error: Update not found"

Kurt Lang wrote:

Any other full version installer I try to download simply returns, Install failed with error: Update not found

Yes, I'm aware. :)

Oh, fer cryin' out loud! Yes, you already said the same thing.

Been fighting a cold for the past 12 days. I'm awake, but not really awake.

Kurt Lang wrote:

Been fighting a cold for the past 12 days.

I hope you win. :)