Music Libraries Fragmented Across Multiple Platforms - How To Consolidate and Fix?

In the beginning there were a succession of iMacs using iTunes of course, and all was as well as I needed them to be in that era. Since then, iTunes has morphed into Music and there are more platforms in the mix, with a mishmash of results. After being more or less forced into Music for a while, and then deciding most of what I was listening to was stuff I already had, I dropped Music some time ago — today the situation is roughly the following, listed in decreasing order of 'completeness':

Mid-2011 iMac (most current OS and app versions) - complete original iTunes Library afaik

iPhone 12 Mini (17.4.1) - Roughly 70-75% of original content, including all iTunes Purchased plus a lot of my own stuff, with no rhyme nor reason I can see to explain what is and is not included.

iPad 9G (17.4.1) - all iTunes Purchased plus a random smattering of my additional content, almost randomly selected. For common artists, some albums included, others not, etc.

M1 iMac (12.7.1) - Originally and for some time, nothing at all - was I guess expecting for it all to happen just automatically (remember "It just works"?) but it never did. Eventually discovered how to import from iTunes but all that showed up from that was the iTunes Purchased content, i.e., well less than half of what is still sitting in the original ITunes music library.

What I want to do now is to fold Music back in, that being the only way I can see to fully populate the libraries of all the platforms, but days of reading here and elsewhere hasn't helped enough. Start with backing up the original library for sure, I get that, and have a drive I can use for that too. But, then what?

I came across a blurb somewhere which essentially said that the iTunes library in the platform upgrading to Music would be shared across all other platforms, which sound good if true. Is it, though? All I have to do is go onto the older iMac and fire up Music from there? And, if not then what are the various hoops that I am going to have to jump through to get this done?

Thanks for any help you can give me and getting me pointed in the right directions before I start flailing around all over the place.