You seem to have a belief that emotion and reason are incompatible. And a fantasy that problems get fixed in the real world without anybody making emotional appeals or having feelings about the problems.
As to the first, a good place to start is Damasio's "Decartes' Error": https://en.wikipedia.org/wiki/Descartes%27_Error
As to the latter, I don't know what to tell you, but it's obviously not the case, and believing otherwise is not any sort of intellectual or rational position. Which is obvious from your writing; you're clearly having a lot of feelings here.
I'd add that it's perfectly possible to critique the conclusions of somebody who's making an emotional argument. The basic point here is that we should have more and better health services. You could say, "This story is heartbreaking, and I feel for her and her kids, but here's why more money is not the way to solve the problem."
I think the actual problem here is that you don't want to "feel for her and her kids", which, yes, is going to make you look unempathetic and disparaging in your attempts to get her to hide her pain.
I agree and will state that whenever a thread like this is on HN you have to walk on eggshells. The minute mental illness or any sensitive topic (hardship, health, whatever is presented) all the compassion comes out from possibly the same people who would otherwise jump all over someone's point of view about a non compassion topic. It's kind of the HN version of 'we won't be angry at you on your birthday day after back to being a dick'.
I also love the predictable 'get help' ok great thanks I am sure the person writing doesn't know hasn't thought of or hasn't tried that already.