OK I owe SeedVault and it's developers a big APOLOGY! I solved my backup problem, but in a way I don't think I've seen mentioned. And it's really simple.
Seems SeedVault works fine but doesn't store it's backups in the folder on the phone marked "Backup". Way too obvious right? I connected the phone to my computer and found that there's another folder marked ".SeedVaultAndroidBackup". Who would know?
Anyways, there were lots of backups there from what I thought was my fruitless effort yesterday. I selected the latest one, stored it on my computer, and then hooked up my new phone. I put the backup file into the ".SeedVaultAndroidBackup" folder there. Then the backup worked just like Harry Potter himself waved the wand.
Seems the internal backup isn't useless at all, just you have to manually move the file to another device. Perhaps someone with Hermione-level skills could organise to have it happen automatically. For most of us Weaseleys we'll just tell our calendar app to remind us every week I guess.
For those muggles who are even fresher and greener than I: to transfer files you have to give permission in the phone. It's a great security feature. When you have the phone hooked up to the computer with a USB cable, go to Settings: Connected devices: USB. You'll arrive at USB preferences. Check the box marked "File transfer". Then your computer's file programme will be able to see and work with whatever is in the phone. Now I think of it, this may be the whole reason I couldn't get the backups to go to USB. The instructions are written by IT guys who take it for granted that you have read the entire manual and know all this stuff like the back of your hand!
Enough already: I hope this helps someone.