I'd probably just buy a cheap prebuilt PC with a big case, or surplus enterprise tower. You can get cheap LSI PCI cards and do miniSAS to SATA if needed.

The biggest concern I'd have with USB is power delivery to the hard drives, but I haven't even done the napkin math so maybe it's fine. The SSDs seem like they might be a waste of money. USB hard drives have a poor reputation, but I don't have a ton of experience to say how much of that is deserved. On a practical level, I'd also be concerned about knocking the cable out.