Private healthcare can become too inundated by need to help everybody too. If that happened they'd be picking and choosing based on who has the most money. If there's more need than availability of care during a disaster somebody is going to get screwed. It happens in war sometimes. Some horrifying shit goes down and limited medical personnel have to make heartbreaking decisions about who is worth trying to save and who should be left to die.
I get triage. I have been on the side of the road in turnout gear at a major accident. One patient per helo, two helos in the State. (We have one, inbound). Four badly messed up victims, and the decision was which one goes out on the bird. Obviously someone who is messed up enough they aren't likely to be stabilized in the local ER, but they have to be in good enough shape to make it to Shocktrauma (about 20 minutes to get in the door). The other three will go to the nearest hospital, 30 minutes by meatwagon, and they've been notified, but it's a holiday and they aren't set up for three critical trauma patients at once.
You gotta make a call, and it is likely you will lose at least one of the four because of it.
Most critical: immediate ground transport to nearest hospital. Maybe they can get that one stabilized before the next most critical gets into Shocktrauma.
Next most critical on the helo.
The other two go out on the inbound ambulances, to the nearest hospital who will hopefully have made progress on or called the first patient in, freeing up some staff to work on them.
The first off the scene may not make that hospital, but if they don't, it was the right call, because they likely would not have survived the helo ride either.
It is a brutal business, triage, but the idea is to save as many lives as possible.
The reality is that there will always be events which, for a short but critical time, will overtax whatever existing capacity is present. That capacity is based on more ordinary times, with some 'overflow' built in for crisis, but the economic reality is that staffing and maintaining facilities at the level of the highest likely crisis means that on ordinary days there will be people standing around empty beds with nothing to do, and that's expensive. It is not economically feasible to be completely ready for the worst case scenario.
At the time I was in Fire/EMS, there were three ambulances within 15 minutes of the scene of the accident I described, with EMT level crews, and two fire companies to respond, at about the same distance. The MD state police had two helicopters set up for emergency transport of patients in the state.
You work with what you have. You do the best you can with that, but reality is that some will not survive.
Keep in mind that in any socialized system, the levels of staffing and preparedness at any given location will not necessarily be decided on rational projections of need, but on the basis of government staffers who might be more inclined to cater to votes rather than patients who are seen as less than influential.