So it seems that as of iOS 13, the device has to be supervised by the macOS device to take Home Screen management instructions.
I wiped my device and found that in Supervision, the Home Screen modifications work perfectly.
But in order to keep my iPhone data, I want to be enter Supervision AND have my iPhone's data in place.
But (!), in order to switch Supervision on without starting afresh on your iOS device, you not only need to backup/restore your device, you need a temporary second physical device too!!! Yes, you read that correctly.
Why? Because Configurator will only enrol the device into supervision while it's having data restored if the backup comes from a different physical device than the one you're restoring to.
So the temporary device is used to basically clone your backup but with a different hardware identifier. Then, that backup can be restored to your actual phone with Supervision in play too.
https://help.mobilock.in/article/w830ax76x0-supervise-i-os-devices-without-data-loss-backup-supervise
I spent 6 hours on this so far. Quite an extraordinary situation.
So, next experiment will be to get a temporary iPhone (with enough space) to run this process on and will report back.