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No, the at fault driver and/or their insurance would be responsible for paying their medical costs. The person who was hit doesn't pay their medical bills and then ask the insurance company for reimbursement. They file a claim with the insurance company to pay their medical bills directly. AFAIK all US states require bodily injury coverage to work that way.
Of course if you don’t have insurance you’re kind of screwed, or your insurance doesn’t cover the full recovery costs you have to pay the medical bills and get stuck with them if the responsible party is under insured and lacks assets.
Accident created expenses seem like one of those things where if someone else is responsible for the costs you should be able to simply transfer all subsequent costs to them, rather than being stuck with bankruptcy or life long debt if they can’t afford to repay you.
Honestly there should be a transitive debt mechanism - but companies won’t like that because currently they can just force people to settle for some minimal payout knowing that they aren’t on the hook for anything that comes up down the road.