Your analogy has cause and effect reversed. The correct analogy is that if I know there is a reward for telling people their car doors are open, I will go around town specifically looking for open car doors. With no reward, I'm just going to go about my life not even looking at car doors.

Your service may be too small for this, but a company like Amazon typically saves money overall by running a bug bounty program because uncaught bugs can be extremely expensive.