Accident took place in Irving, TX. I had the same question and some quick research found this:
"Texas law requires every driver to carry minimum liability insurance coverage. Proof of auto insurance is required for car inspection, registration and some driver license transactions. The fine for driving without insurance can be up to $350."
Pretty much the state can punish the driver for driving without insurance, but can't force them to pay for anything without a civil lawsuit against them from the courts. And if the person is already broke and has no assets, then it's unlikely the plaintiff would really get anything worthwhile in return. The best you can then hope for, in spite, is to force them into bankruptcy and take anything they actually have that is of value that isn't necessary for their livelihood. And force them to never drive again until they have insurance, which the state may already take of....