Well if the Troopers are willing to pay for that ambulance ride...
Seriously, this situation is such a perfect summary for how broken the US is. Life or death emergency? Well you should fucking wait (hmm, isn't wait times one of the claims for why America is superior in health care by brains for shit people eager to defend rampant unbounded greedy capitalism?), who cares if you or a loved one might die waiting.
There's been a few times I took family to the emergency room in the last 3 years. I drove them because it was always going to be faster than an ambulance, cost wasn't even a concern at the time (but it is a relief afterward seeing how much others are getting charged for rides). And that's not even with a barely functioning system like it sounds like Little Rock's 911 system is. (Maybe someone should ask their governor why their 911 system sucks so bad?)
Which, sure, emergencies shouldn't be carte blanche to risk further harm, but likewise speeding shouldn't give cops that either.
Something to keep in mind, the closest hospital may not be the best hospital for some emergencies. If I have someone experiencing symptom of a stroke I do two part assessment. One is basically a presumptive stroke diagnosis, the second is a severity scale. If there is a high level of severity, I will bypass the closest ED and get them to a specialty stroke center that can offer additional treatments for large vessel occlusions. Most decent sized hospitals can give medications to dissolve a clot for strokes, but typically only large specialty hospitals can go actually remove the clot with a catheter. Patient outcomes are significantly better if they can quickly be delivered to a specialty center with their teams being activated prior to arrival. Even on a smaller level stroke being seen in a local ED we can have them clear a CT room to expedite treatment of them on arrival, and have their stroke team standing by for us.
For someone having chest pain, I can do an initial 12 lead EKG, and if they are showing signs of active heart attack (STEMI) on the EKG I transmit it electronically to the ED and have their cardiac catheterization lab either opened up if it is off hours, or a bed made available in it for an emergency case even if they are already open. Cardiac cath teams can often be off site, but on call at night, and based on my EKG they can be started in before the patient even gets to the ED.
There's also specialty referral centers for eye injuries, severe hand/foot injuries, burn injuries, major trauma, high risk maternity, etc that might be better seen elsewhere.
We also track the number of ambulances waiting to offload patients, and their overall ability to accept patients. Sometimes going to a hospital further away can actually get someone seen by a doctor sooner, if we go to a hospital with lower patient wait times.
Unfortunately, not all EMS systems are adequately funded and staffed and a number of systems are seeing units being held up in the hospital waiting for beds, preventing them from being available for additional calls. So there may be cases where the quickest way to get them to appropriate care is a private vehicle.
I do think it was a bit risky having a teenage driver try to get their parent to a hospital though. Relatively inexperienced driver in a high stress situation, driving at higher than other traffic speeds is a recipe for an accident, but the Trooper seemed very quick to PIT them as he approached. Not sure if there is more video from Little Rock PD prior showing more of an actual pursuit. Performing a PIT on an exit ramp is safer than on the highway itself due to the lack of other traffic and lower speed, but doing it right near a hospital seems more of a case of can they do it, vs should they do it.
And I am not sure of their guidelines, but if they truly cared about the mother, one of the other officers on scene could have very quickly given her a ride to the ED entrance, while the rest remained to sort out the driving situation. Uncuffing an 18 year old that just went through that and having them drive the mother seems like a less than ideal choice.