Just like a car, you can turn on the headlights and they are nice and strong, but you can't start the car.
A bad connection may pass a low amperage draw, the headlights, but not a high one like a starter. Somewhere in your system, you have a bad connection. When it happened to me I just ran new wires directly from the switch to the starter and cut out the run through the key and harness. Sure I can now start the engine without the key, but I don't because I know that if I do, I might burn out a diode in the alternator.
Most often though, it is the engine ground which is at fault. A bad connection, a bad crimp on a lug, or too many wires on the same bolt. Especially on a boat in saltwater, a pinhole or cut in an out of sight place in a wire can corrode a wire enough to break the connection over time. These are particularly hard to find as they can be behind paneling or in a harness.
All that makes sense.
Unfortunately, it leaves me twisting in the wind on the topic of "what is the right way to fix this? I suppose I could start by cleaning all the connections and seeing if the start button will work again. If so, that would definitely answer the question.
I'm surprised that the solenoid would need to pull enough amps for this to be an issue. I thought that the entire point of a solenoid is to use a tiny amperage to control a much larger amperage. I would expect that even a small amperage through the button would be sufficient to close the solenoid. This is one of the things I HATE about the Internet: searching for this part returns a jillion vendors who want to sell it to me, but I can't find a single spec sheet that specifies the amperage required to close the solenoid.
The "pinhole" comment is interesting/unsettling to me. These old (original) wires are untinned copper, so any intrusion is going to corrode them pretty quickly. What you're suggesting is that a nearly invisible breach in the insulation could be sufficient to cause a lot of corrosion and render the wire too resistive to be useful. Of course, under those circumstances, I'd still see full voltage at the load end, as long as I don't try to pull any serious amperage. Frustrating.