I can add this to my car insurance for a few bucks a month, but I do not see when this would be needed. So, let's say I get into an accident caused by an uninsured driver. Maybe I suffer severe injuries and damage to my car. Wouldn't my health insurance and my own car insurance cover this already? In other words, assuming I have comprehensive health coverage and car insurance, what loss could I possibly suffer other than maybe having to pay some deductibles?
There are many secondary out of pocket costs associated with dealing with a major automobile accident and injury that you will eat if you are in an accident with an uninsured driver. Even in the best of cases people are unlikely to made completely "whole" after a major accident, but with an uninsured driver it is guaranteed.
Depending on how you believe you will be able to handle this situation financially, whether you can live with being on the hook for uncovered secondary expenses, deductibles, co-pays and lost wages, it may make sense to do this.
Maybe I suffer severe injuries
Your health insurance will cover treatment, but you are still paying deductibles, co-pays, hospital transportation, lost wages/time at work, any temporary accommodations you need while recovering, medical equipment, and other expenses not covered by your health insurance
damage to my car
If you carry comprehensive insurance you will likely pay higher premiums going forward from a major claim against yourself. Anything that is not explicitly covered in your insurance contract will be yours to pay, this can be things like impound fees, tow fees, rental car past some point in time, and depends on the terms and additional services you pay for.