Migrating via Time Machine from an old intel running High Sierra to a Mac Studio running Sonoma

Is that external drive in a USB 2.0 case or a USB 3.0 one?

A mechanical hard drive in a USB 3.0 case is limited by its own speed - which would top out at maybe 100 – 150 MB per second, say 10 seconds per GB, or 10,000 seconds (2.8 hours) per TB. USB 2.0 has a nominal transfer speed of 60 MB per second, but 30 – 40 MB per second is more common, so then you'd be talking 8+ hours per TB.

There's probably a bit of extra overhead involved in going through lots of individual files on a Time Machine backup, so if you're migrating from a nearly-full 1 TB USB 2.0 drive, I'd give it a while longer.

As for the blurry background, I remember seeing something similar when I upgraded my. old iMac – possibly when I upgraded it to High Sierra.

Somewhere along the line there was a change, and macOS started to blur the desktop photo when you were logged out. You should get a login prompt at some point – but seeing a blurry image when you're logged out might just be normal, not a sign that the installation is dead.

Although the BEST Rotating magnetic drives can source a single initial Burst of data off the drive at near SATA-3 speeds, most can not maintain that data rate.

The limiting factor is the speed at which the platters spin the data under the read heads on the drive. Then the drive goes completely quiet for a (comparatively) long time while the head is moved to a new track.

Steady-state transfer rates for large files tend to be lower than 50 M Bytes/sec rather than 150 or higher.