Catalina will for sure recognize the software on the classic. I have multiple versions from 5th through 7th generation, and I can get them all working. I'm not sure if the fact that all of my iPods are upgraded with a SSD makes a difference, though there's no reason I can think of that it would. The iPod isn't going to 'know' if it has a typical HDD or an SSD.
My suspicion, and I'm more-or-less totally speculating, based on what I think is a possible reasonable starting point is that the formatting on the iPod is incorrect. In particular, if an iPod was set up, for example, as a FAT32 iPod in Windows, or a Mac for that matter, Catalina won't be able to read it.
You'll note that if you have any 32 bit programs on your Mac after the Catalina upgrade, these will be grayed out, and if I recall correctly (I'm running Mojave again!!!), covered with a circle/slash combo. Apple abandoned support for 32 bit file structures with the Catalina upgrade. Thus if your iPod is set up with such a file structure, I expect it would symptoms similar to what you've described.
The fact that Disk Utility doesn't recognize the iPod is troubling though; however, this could be a function of the larger 32/64 bit issue. Perhaps Disk Utility doesn't recognize any drives formatted in that way... even if that sounds like an insane oversight to me.
With all of that said, if it were me, I wouldn't abandon hope. If none of the things you've attempted worked, I'd probably get a bit more technical. One option would be to remove the drive entirely from the iPod, put it in a caddy and hook up to your computer as a drive. Assuming the drive is functional, you should be able to get it to respond to some computer somewhere. While you have the drive out you could just upgrade to a different drive, which would force the iPod to restore since would have a blank drive. I've actually upgraded a couple of Classics to SSD recently and have had success with this method.
And with all of that said, I'll close by stating that I recognize very few people are as committed to this device as I.