There are other places to buy wiring harnesses that have come highly recommended--wasn't British Wiring (
https://www.britishwiring.com/) one of them?
In any case, I agree about the quality of auto-store wire; it can be pretty bad stuff. I usually solder things when I can, and the insulation on most of that stuff shrinks back when it gets hot! Radio-Shack wire is no better, in fact worse, but I think if you try one of the electronics outlets like Digi-key or Mouser, you'll do OK.
Assuming that a lot of the wiring is in bad shape, and not just a couple wires, I think I'd just buy a harness. Anything to enhance reliability. Electrical problems can just stop you cold--bad thing to happen in the middle of traffic.