Downloaded the Amazon Music app for PC. This was deftly hidden, Amazon's own site links only a web player and if you click "Download App" it gives you a QR code. For your phone I guess. I had to go to the Windows Store.
Although Amazon's help pages claim the PC app can download purchases, I still show no purchased music. I think this is because every time I log in on the US store the app redirects me to Canada. Yes, I am in Canada right now. But I was in the US in 2007. When I bought the music?
Alright so here's the least useful advice you're gonna get today. It is possible to "sync" an Amazon Music USA to an Amazon Music Canada account (and one assumes other combinations of countries as well). The way to do this is to speak to a human customer service representative¹ and request it. Then it takes 48 hours. I'll let you know in 48 hours if it worked or not.
¹ In order to speak to a human Amazon representative, you must: Solve the nefarious labyrinth of Dr. B