It “shuts off” despite spitting water?
1. Does it completely shut off, as in not running at all anymore?
Or
2. It goes into limp mode?
If while it’s idling and running, and is peeing water, how hot is that water coming out the pee hole and how is the the flow?
If it’s peeing water normally, really question if it’s actually shutting down from getting to hot or if there isn’t a wiring or ignition issue somewhere?
Replacing an impeller is not a bad idea, but it “could” be working properly.
Sounds like the owner is relaying info to you and that’s tough to go off of for a 3rd party to accurately diagnose. And what he’s telling you is his interpretation of the symptoms, which could be totally inaccurate. Not by his fault but just not knowing outboards well enough.
Good luck with it.