The statement that "I know the iMac is older but it works perfectly, so I expect full downwards compatibility." is a bit unrealistic. There comes a point at which an older Mac that "works perfectly" will not provide the full compatibility with newer versions of iOS that you seek.
A few years ago, Apple supported manual synchronization using iTunes under El Capitan or Sierra. Then they raised the requirements, and now they don't. Catalina should still be good enough to do a manual synchronization - but if your iMac is still "working perfectly" five years from now, that doesn't automatically mean that Catalina will be good enough to manage an iPhone running iOS 32.
I'm not surprised that attempts to establish a Bluetooth connection failed. Macs and iPhones do not work together in a pairing way. It is possible to carry out manual synchronization over Wi-Fi, but only if you first synchronize your Mac and iPhone over a wired connection and check off an option to allow the use of Wi-Fi in the future.
Use the Finder to sync your iPhone, iPad, or iPod touch with your Mac - Apple Support
As for troubleshooting the USB connection,
- Check that Finder > Settings… (Sidebar tab) has a checkmark by [X] CDs, DVDs, and iOS Devices.
- Check that the USB cable(s) and adapter(s) you are using can carry data and are not "charging only" cables. If you use a cable that has only 5V power pins connected, the iPhone may charge off of the Mac - but neither the Mac or the iPhone will see the other.