There are some people who object to insurance on religious grounds; they see it as a form of gambling. They are not wrong.
Just like any bookie or Vegas casino, the house is calculating the odds, planning on paying out a certain percentage of the take, but planning to make money overall and adjusting the odds to suit the bet. If a bet doesn't look profitable to them, they won't take it. If it looks like someone is not playing by their rules, they'll keep all the money and kick them out of the game.
I don't know the situation right now, but when my kids were your age it was common for insurance policies to have a provision that all drivers be over 25. If anyone was under 25, then some companies would charge an extra premium to cover their perceived extra risk and other companies would simply refuse the bet (the same was true of male vs female drivers too). When challenged under Human Rights laws against age (or sex) discrimination, the bookies, er, actuaries trotted out their charts and figures to prove it wasn't discrimination, it was simply a fact of life.
Virtually anything and anyone can be insured. If a pretty actress has nice legs, some insurance company will bet nothing will happen to them and some bookie will offer odds something will. Dad (or Mom, or whoever is running things at your family company) can shop around and decide if the extra premiums are more or less than hiring a driver that meets the policy provisions or shuttling an acceptable driver back and forth. If they are caught putting a ringer (you) in the game, the insurance company will void the insurance and keep the money. Oh, and tell all the other casinos, er, insurers that they are known cheaters so they can't play anywhere else either.