If my imaginary four-year-old has a fever in the middle of the night, I first phone the nurse-on-duty, and if they determine I need to go somewhere, I see if there's a nearby shared e-vehicle (and in the middle of the night, it will sure be), and use that. Or, like, call an Uber, which
might be slightly faster, but for a fever, well...
And if the situation is truly urgent, they'll send an ambulance, which should take less than 15 minutes, and will offer the best care available in that time window anyway.
And I'm absolutely sure the situation is entirely different where you live, but isn't that the real problem then?