If someone injures you, and your insurance doesn't cover it, can't you sue them? If you do and they can't afford it, can you put them into debt? Seems reasonable rather than you going into debt for medical bills?
And if they move to another state, have fun transferring your judgement over there and trying to find their bank accounts and employer(s). It's like getting blood from a stone. It's easier and cheaper to just pay a tiny bit more in insurance premiums.
Once you have any sort of reasonable income you should strongly consider buying significantly more insurance than the legal minimums.
https://www.allstate.com/resources/car-insurance/uninsured-m...
No effort was put into finding them, even though there are cameras at every intersection. Through my own effort I got a grainy photo from a local business, but not good enough to see the plate. Until the police actual police and there are consequences for people, things will only get worse.
You may desire to transfer the debt onto them, but the hospital will absolutely not accept that. Why would they? If a bank owed you $1 million, and they said "oh, now Jim owes you that money, not us" you would similarly refuse.