This could drive up the price of full coverage (i.e. uninsured/underinsured motorist insurance), but you're not required to buy that. In principle it should also reduce the moral hazard of insurance -- if people have less insurance then they have to be more careful about causing damage because if they do it comes out of their own pocket instead of the insurance company's, or puts them at risk of being charged with a hit and run if they flee.
The actual problem is that too many people don't own anything anymore, because otherwise they would want to carry insurance to protect their assets. Leave too many people poor and desperate and you can see what happens.
It should be easy for people to grok that by having a large number of uninsured motorists on the road, their own insurance premiums are increasing. Thus, people should be petitioning the police to catch more uninsured motorists. Why isn't that happening?
https://www.cbsnews.com/miami/news/investigation-cops-regula...
But they don't need to do insurance spot checks - in my state they run plates of cars that are in view and, presumably, they are alerted if you don't have insurance.