Thanks for the replies and helpful info. We want to keep the original look of the bike. Other than a new Brooks saddle, new tires, brake pads, and cork grips, it's completely original at this point. It's our quick errand bike to pick up dinner, a 6-pack, etc, so the kickstand is quite useful.
I looked more closely and it's definitely the pin that broke. The spring is still attached, totally gunked up with grease. If I can't find a replacement pin at a hardware store, I may just take it to the LBS and see what they can do.