Out of curiosity, which mechanism are you thinking about that would lead you to the conclusion that hospital capacity doesn't impact mortality? The story that I have heard the most is that hospitals start to turn away people if they don't have beds from them, leading to either delay in care or no care at all. Both delay or no care lead to higher mortality. I cannot really think of a story where an over-capacity hospital leads to the same or reduced mortality.
If you believe that hospital closures lead to hospital overcapacity or delays in care, you can find some literature that shows that closing hospitals leads to an increase in mortality in the community [1].
First, it's that hospitals effect on covid mortality is lower than believed. Second, while morbid, I assume mortality risk can be assessed pretty accurately by hospital staff. If you must triage, triaging someone who has a 99% chance of dying in the hospital and a 100% chance of dying outside the hospital raises the expected deaths by 0.01. Hospitals tend to serve everyone they can (for good reason) but the value add part comes from serving people who have a high chance of dying out of the hospital AND a relatively low chance of dying in the hospital.
These could be wildly wrong assumptions.
I would think hospital closures would cause a large increase in mortality rates as you don't get to triage effectively at that point.
I think the emphasis on ventilators at the beginning of the pandemic likely caused an increase in mortality in hospitalized patients. At peak panic, there was a big push to create and distribute ventilators, including open source projects, big name tech billionaires dedicating resources, etc. Then we realized these were actually damaging lungs further, and nixed that treatment, switching to oxygen supplementation only.
One of the regions got an alpha variant wave sooner and the local hospitals actually went over capacity, with triage going on. You can see it in the time-variable CFR plots.
https://www.linkedin.com/pulse/what-cfr-data-can-tell-us-pan...
Roughly it seems case fatality rate increased almost 2-fold, despite hundreds of patients being transported out of the region by ambulances.
( https://translate.google.com/translate?sl=auto&tl=en&u=https...)
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7685049/
There was a lot of misinformation and confusing data at the start of the pandemic regarding hospital beds. Here in the US, hospitals tend to operate near capacity as SOP - a result of for profit healthcare seeking max profitability. Most hospitals here also started reserving a percentage of their beds, both ICU and general care, for covid patients. These reserved but sometimes unoccupied beds were not factored in when local hospitals reported being at capacity - so a local radio station might say "these local hospitals are at max capacity" without reporting that they still have 5% beds free specifically for covid patients. I suspect this may have caused some deaths by people not seeking care because they thought it wasn't available when it was, however I haven't seen anything to validate that hypothesis (and I doubt there will ever be any data).
One interesting sidenote is that the military hospitals that President Trump rolled out while he was still in charge had very little effect:
https://www.npr.org/2020/05/07/851712311/u-s-field-hospitals...
I've always suspected claims that overcapacity hospitals drastically increasing the mortality rate are not supported by data. Obviously if more people have covid, more people will die from covid, but normalized to the number of people that had the disease.