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Incidentally, it's also the same story for colorectal cancer, the 2nd most common type of cancer. Here's another study on the topic. [1] They have a survival rate of 88.6 vs 85.9 for breast cancer, but it's a large observational study that's not normalized, so the confounders/biases there probably explain the reduction in survival rate at non-NIC hospitals. Colorectal cancer is even smaller - 0.2%.
NIC hospitals only showed a significant effect on cancers with low survival rates, and especially on rarer cancers. For instance with pancreatic cancer 93.8% of people who went to a non-NIC hospital were dead in 5 years, by contrast 'only' 87.5% of NIC hospital patients were. Feel free to look up the data yourself. I'm not searching for cherry picked studies, there are none - as there seem to be oddly few studies on this question, and they all say the same thing. What benefit there is is quite small, and heavily driven by extremely rare things.
The claim: For the overwhelming majority of things people to go to the hospital for, where you go doesn't really matter.
You win, as always.
If your local hospital can treat e.g. colorectal or breast cancer to the same degree as a specialized institution, then they can certainly competently treat the overwhelming majority of other issues that people show up to the hospital with, which are generally going to be substantially more mundane with rather more 'commoditized' treatment available.