I feel like I am the only person around me who maintains (changes oil, rebuilt carb etc.) equipment and people seem to treat lawnmowers the worst. I swear I think people throw them away because they're out of fuel sometimes.

Unused working lawnmowers can rust and stop working.

Lawnmowers rust if they're not properly maintained, regardless of use. There's no evidence that shared lawnmowers would rust slower. Probably the opposite because when people own their own lawnmowers they're move like to maintain them properly. So no, sharing wouldn't reduce the net demand.