The bizarre thing is I can't find bare drives cheaper than that on Amazon - how are they making a profit on these?

Generally speaking the drives in external enclosures are from the bottom slice of the still-viable drive pool. You'll often find that they have shorter/less comprehensive warranties.

Larger market of average, undemanding home users for external hard drives.

Smaller market of more demanding enthusiast users for internal drives. (Larger businesses aren't sourcing drives from Amazon or Newegg, unless the company is called Backblaze.)

Latter costs more to support and as niche users generally pay more. (The more price conscious resort to shucking external drives.)

At least, that's how it's been explained to me. I don't know if that's the entire story though it sounds reasonable enough.

Maybe it's one of those drives that comes with a demo of the backup software and they're hoping you'll buy the full version?