I was unable to solve this, so I've just moved the HDD back to the older USB enclosure, where it works great.
I think it could be a hardware or driver issue. Looks like different USB 3.0 to SATA chipsets have slightly different characteristics, which result in incompatible file systems when the HDD is moved between the USB enclosures :-(
Update: since that time, I've bought a new desktop and connected that HDD directly to SATA without USB. The file system was recognized OK.