With a turn signal-equipped bike that has an unknown history, plus obvious signs of ham-fisted, kludge, repairs this can seem overwhelmingly complex. IMHO, best to take it in small, manageable, steps. The one thing that immediately grabbed my attention was the new keyswitch. They exist in multiple flavors and are not universally interchangeable. I'd start there...multimeter in hand, schematic within eyeshot. Verify the switching function of each key position and each lead at the harness connector. There may be a wiring mismatch, despite the connector shell halves fitting.
Next up, the circuits that are powered-up as soon as the key is rotated to the "on" position: brake light, horn, neutral light, turn signals.. Could be something as simple as a single reversed pair of bullet connectors.This includes the rectifier, HL, speedo, TL...looking for one that might be reversed (+) for (-), or shorted...like the horn or tail light. There could be a short caused by an insulation rub-through...4 turn signals and the rear brake switch pigtail are obvious places to look.
Compared to a K0-K2 model, w/o signals and most of the handlebar switching controls, there's significantly more things that can go wrong. That said, even a `90s model electrical system is still quite basic...a.k.a. simple. The "nuclear option" is pulling the wire harness and verifying every lead, on the bench...unwrapping it, if you find a problem, or want to be extra-thorough. They rarely have defects but, rarely isn't the same as never. And, that'd allow you to solder the splices...which are simply twisted & taped. Still no luck...pull the remainder of the electrical system, then verify & check every lead & connection; that's a lot easier on the bench than it is in-situ, i.e. on the bike.