The very first thing I would do is invest in a Honda Specific Service Manual. That way you can start tracing the circuit that makes the ignition circuit do it's thing.
Because of the age. I would check things like the coil wiring making sure the wires are not only attached, but the connections are tight. Through the years these wires move, and the strands get weak and break off.
Also go through the Start/Stop switch, the Ignition Switch, and any switch or part that has wires running to it. You can check them one at a time, and fix them one at a time. This way you don't have a bunch of wires off of what they go to, and can't remember where they went.
But do the manual first. Buy a used Manual specifically for the year and model of your ATV. Don't buy Clymers or any generic type manual. Buy the HONDA Specific.