Sounds so familiar. This problem first showed up on my (now) 100k 1995 FLSTC over three years ago on a trip. Bought a new solenoid, starter relay, battery, none of which helped. Over the years had it looked at by several dealers to no avail. Finally just carried a length of 12 awg wire with me to shunt from ignition relay to starter solenoid (green wire), and that seemed to get me started.
Usually, once it was running and I shut it down after riding some, I could get it started, but still often had to use shunt wire. Only thing that saved me during this time was that I had long before rewired my ignition switch so that I could run without the headlight, so I could start without that drain on my battery.
Finally got so bad even the shunt wouldn't work. Checked the voltage on the battery (another brand new one) and discovered it wasn't fully charged. The light turned on in my head and I thought maybe it's a charging problem.
Did the normal things of checking voltages at various points in the starter circuit and this verified that my starter wiring was OK. Then did a load test on the charging circuit, which indicated I wasn't charging.
Checked the voltage regulator and it was OK, so then checked the alternator stator. It was grounded, which it should not be. Replaced the stator and it now starts like it was brand new.
Here's how to check your stator. Unplug the voltage regulator from the stator. There's a two-conductor wire that runs from the regulator to the front of the engine case on the left side of engine. Unplug that plug.
First, measure the resistance between the two terminals in the socket (on the engine case). It should be low, but there should be some resistance, not quite zero. This indicates that your stator wiring is not broken. An open circuit here means you need to replace the stator.
If this checks out OK, then measure the resistance between each of the two conductors in the socket and ground. In both cases, this should be an open circuit (infinite resistance). If either shows low or no resistance, your stator is grounding and needs to be replaced.
Hope this helps,
Ken