No need to check the handlebar switch - if the solenoid clicks at all, it got what it needed from the switch.
A solenoid is a solenoid, so lots of other stuff should be able to be made to work without too much effort - junkyard, AutoZone, Pep Boys, etc.
I've also read of disassembling similar solenoids on one or two other sites and giving the main internal contacts a bit of cleaning (sandpaper) and being good to go. Never done it myself, but I think I'd start looking at that solenoid for how to take it apart.
Meanwhile, carry that large screwdriver with you (or however you can jumper those two wires together once you've taken the solenoid out for refurb) and keep commuting!