I have had freezes, random restarts, etc. and basically gave up and just wiped the system clean. Since then I no longer get the restarts but I get the occasional split second black screen as if a screenshot was being taken.
Doesn't sound video card related, on the surface. Could be that there's a bad cable leading to the backlight, though - do you find that slowly tilting the display hinge back and forth does anything?
I am also keeping my machine off things like Adobe Flash just to keep the graphics lite, but maybe I should just go back to making it work its normal load?
No, you should obviously do your best to make sure the system never encounters any problems until the repair program ends. Certainly don't stress test it or anything like that.
Funny thing is that even when I reboot in one of those safe/reinstall modes where the "Welcome" message scrolls along in various languages, there is clear distortion in the text there.
Those're compression artifacts, I'd say. The quality of the video itself is more likely to be at fault than anything else in that particular case.
Is there any test I can run on my end to check out the video card(s) fully on my Early 2013 MBP 15"?
Try and get the system to fall over under "normal operations". Use iPhoto (or Photos, whatever), play around in iMovie, play high-res videos in iTunes / Safari. Keep a camera on hand so that if you experience the sort of display corruption indicating card failure, you can take a photo / video of it.