If you cause an accident and someone's car is destroyed, your insurance company isn't going to pay,
Except that's not actually true. Provided a policy is in place the insurance company is, by law, liable for third party claims irrespective of whether the policyholder has broken any terms of the insurance contract. They are able to claim any losses back from the policyholder though.
According to this site, the insurance company will usually try to direct you towards the other driver until you lawyer up threaten to take them to court.
So amend my statement to - isn't going to pay unless you shell out for a lawyer to make the relevant legal noises. Assuming you know that you have that option. And assuming you can afford one. Tough **** if you can't.
Theoretically, what happens if you hit a pedestrian in that uninsured, unlicensed situation? And if they receive life changing injuries?
The insurance company is obliged by law, they can't legally evade payment for third party damages. The insurance ombudsman will ensure that this happens if the insurance company don't play ball.