I had the same issue with my '04 C-model back in June '05. Several things to check.
Question - did your Trip-A & Trip-B odometers reset back to "0" ??
1.) Check that all battery cables are clean and tight (with 10mm wrench or ¼-drive ratchet and socket). Even check the connections at the starter relay - they can also work loose.
2.) It's not so rare - look at the RUN/STOP switch contacts inside the switch housing. They can get dirty, corroded or burnt and loose contact. This circuit feeds juice to your coils. While you're in there, also check and clean the START and headlight cut-out switch contacts. These have no immediate impact on your problem, but do this as a precaution as long as you're in there cleaning.
3.) Check your front coil at the left side frame rail under the gas tank. Due to the design of the bike, the tank acts as a big "water funnel" and allows the mounting bolts for the front coil to rust at the frame mounting tabs. This causes the front coil to loose its ground connection and the bike will die as described.
4.) The final area is the most tricky and took me a while to find before I joined any VTX forums. Bareass172 has documented this well at https://www.vtxoa.com/forums/showthread.php?t=194653
The reason the bike started when jumped at the solenoid, is that you weren't sitting in the seat. Doing so puts pressure on the ECM/ICM harness from the bottom of the seat pan and under many circumstances can cause the power to the ECM/ICM to be cut by the break inside the insulation for the black wire (power feed) or light green (timer pulse feed wire from the pick-up coil)
You'll find your problem in at least one of those areas.
Let us know what you find.