Am I seriously going to have to keep this clunky old infernal machine running for weeks upon weeks to sync the blockchain?
It sound’s like the bottleneck is I/O on your hard drive. A couple of suggestions:

Copy the data directory from you other machine which you have already synced, this will skip you the initial sync entirely.

A lot of read/write is happening in the chainstate directory and full disk encryption probably doesn’t help. You can move the chainstate directory (around 3GB) to an SSD. If your computer doesn't have an SSD you can use an SSD thumb drive. There are instructions on how to do that here