Welp, solved it already. Had to dig about 10 discussions deep in the Apple forum Google results, but someone else had the same exact scenario.

Turns out that when they did the reset, the image they put back on the laptop was the original Sierra, and the backup was Mojave. It looks like Migration Assistant doesn't like that the backup was for a later version of MacOS, so it just sat there doing nothing.

Instead, I booted into recovery and the restore began immediately. Others who weren't able to do it that way just set up the MacBook, updated to Mojave, then restored.