That generates more supply (mechanics) not less demand (volume of broken cars). Sometimes having excess demand is ideal to keep the market balanced in your favor.
Increasing price would just move up the demand curve (less people willing to spend the increased amount on fixing their broken car) and the mechanic would earn more.
ofc overtime it's likely more mechanics enter the market to compete, but that wouldn't be instant. and when it does happen the table stakes would be that everyone's phone call get's answered