Really late start to my day, but so far it's going great! I pulled the starter clutch out again and looked over what I'm working with... I didn't notice before but two of the three steel rollers were sticking or just not sliding smoothly. I must have got them swapped around with the spare parts because digging through my bag I found two that were bright and shiny. Cleaned those and dropped them into the clutch, and they appeared to slide smoothly by hand. I put everything back together and gave it a shot... several cranks of the starter and not a single slip! Hot damn, I'm back in business!
Interestingly, I also did not have any problems with the starter continuing to crank after I released the button. I'll have to keep an eye on that because it may start sticking again once it gets hot, in which case I'll swap out that solenoid with the last one I have. The important thing now though is that I can reliably crank the engine, so I can get back on to troubleshooting the idle problems.
Thanks again everyone for all the advice!