Sounds like a sticky issue; but it also sounds like you're doing all the correct diagnostics. I haven't had stock wiring on any of my last 3 or 4 cars; so I won't be any help there. Replacing those old OEM relays with modern components is a major improvement, so job well-done there.
One suggestion I can offer is to check your starter (and solenoid). Really strange behavior, from your description. Maybe the solenoid is failing, or the bendix gear, or???
It would be interesting to know what the problem really was, if you don't mind posting your results here once you've solved the issue(s).
Cheers, and welcome to the forum!! 👍