Kenmore has a few models that are known to have faulty bobbin winders. Some were made out of pot metal and are very prone to breakage and not working correctly. I don't know if this is one of the models, but the winder could be pot metal, so don't force it. YOu'll need a new tire for sure, but I'd just start with Triflow and then maybe a penetrating oil if that doesn't work. The objective is to get it working without busting it in pieces. Ask how I know this. I do think there are enough donor machines around and perhaps this winder is still available for purchase, but you'd find more info. about it on the vintageKenmore yahoo group.
I would bet that a penetrating oil might get things going for you again.