Berk, without seeing your starter linkage I'm not sure if it's rigged OK or not. The starter lever should not touch the pinion at all in the at-rest position, you should have about 1/16" clearance. The lever should push the pinion into the accesory case 7/16" before it pushes on the starter button,total pinion travel should be 9/16" (another 1/8" after button is engaged). You should also have a return spring on the linkage.
When you re-rigged things after the starter cable broke, I'm wondering if that caused the starter clutch to continually engage the crankshaft gear, and caused some sort of failure. Normally the engine overruns the starter after the engine starts but only until the starter knob is released-- in other words, for just a second or so at each engine start. Continual engagement (not designed for) could bring on a failure. I can't see how you could have just pulled the lever closer to the broken cable & reattached it & still have it adjusted properly. I had a similar cable breakage, but reconnected the starter lever by using a sort of floating link to extend the broken cable. I was very careful to get the linkage adjusted right, per installation instructions by Canadian Aero Mfg who made the starter clutch.
I'd be pretty leery of running the engine any more before checking it out further. I'm with Dave, I'd pull the starter & check out the gear teeth & see if the starter clutch seems to be malfunctioning. Hopefully if the starter clutch is gunnysack it hasn't come apart or chewed up any gear teeth.
Zane, the bendix system you descibe is pretty much what Lycomings use, ring gear & all (mounted behind the propeller). The Continental system is naturally superior (like everything else Continental

) but does require proper adjustment. I think lots of people get in trouble by not replacing the starter clutch at engine overhaul time ("ah, it's working OK"), then when it fails later they're surprised.
Eric