Yes, I do need a license to operate a vehicle on the road. But, I actually don't need to buy my own insurance, because I live in New Zealand. Here, we have socialised insurance which covers accidents in general. Vehicle operating fees (handwave) do vary a bit, basically depending on the cost of that type of vehicle to society. We also have 6-monthly or annual safety checks on vehicles (depending on age of vehicle AFAIK; I still haven't owned one newer than mid-90s), if they are to be used on the road. And, we have a thriving car culture, despite my best efforts to get everyone moving on bus/foot/bike, but that's a distraction to the current discussion...
When you earn that license to operate your car, part of the deal is that you shouldn't operate your car if you think it might not be safe. That requirement does not go away depending on who fixed it last, or for that matter, the car's entire history.
I think there are some serious issues with regulating people's repairs (or just tinkering with) of their own stuff. For a start; I can't imagine a practical implementation of this "car repair license" which would solve any realistic problem, so to me it sounds like a proposal for mostly-useless bureaucracy.
Beyond that, I (and my employer) put a lot of value on the knowledge I've gained to be able to competently, to your example, fix my own brakes (two weekends ago, I replaced a wheel bearing - seems to be working fine). I think that adding barriers to technical work effectively dumbs down society in general. It also encourages waste - if the cost of a repair increases (as it probably would, if the licensing requirements increased), then the broken thing is more likely to be discarded wholesale, rather than repaired.
Maybe we should move on from this car discussion, and consider other regulated machines in common use. Radios, in particular, seem like an interesting parallel - they operate in a public space and carry safety risks. When radio regulation started, radios were similar complexity to concurrent cars, but now are largely software controlled. Do you think you should be legally prevented from changing the driver/firmware in your WiFi radio or cell phone?
