Wore out three sets of tires already!
It was more prevalent in the mid to late 70's from what I've seen.had several newer Schwinns with the repair kit installed.
It looks like when Schwinn changed the angle of the kickstand in the 70's to a more vertical position to solve the stress on the kickstand holder when someone sits on the bike with the kickstand down, it caused another problem. I was wondering why I've never really had any problems like this with my balloon or middleweight bikes. I checked some of those and found they that had as much or more slop than the lightweight I was fixing, but the bikes didn't have any problem staying upright. By moving the kickstand more perpendicular to the ground, the play to the right side will cause the bike to push forward, causing the problem I was having with it not staying upright.
I did get those pins off of ebay and they appear to be an exact match, but I decided not to bother with changing it out right now. It seems to be good enough, but I'm glad I'll have them if I need them in the future.