The argument that you are making has been a named fallacy in economics for over 150 years: https://en.wikipedia.org/wiki/Parable_of_the_broken_window.

But aside from economic theory, everyone who has ever cared for a sick relative should have the evidence to see that it doesn't raise GDP. Yes, you spend money on healthcare. That's good for the doctors you buy from. But you give up time at work, you give up vacations, you cut back elsewhere. Every cent that you spend on that healthcare isn't going to wherever else you would've spent it. This isn't a counter-intuitive finding. It's just common sense.

It’s a cost for the individual but not a cost for the corporation whose cut corners are responsible for the illness. What’s good for profits isn’t necessarily good for health, even if in the big picture and long run what’s good for health is good for the economy as a whole.

Often true. But not relevant to whether it increases or decreases GDP.

Unless it increases GDP in the short term (higher profits) but decreases it in the long term (fewer people to consume goods and drive the economy).

You assume that all money would be constantly spent. Maybe in the US, but in other parts of the world people do save money for these 'unforeseen emergencies', and that money is then 'unlocked'.

In which case, GDP in the US isn't overstated because of the costs of caring for the sick. That's a bit awkward for your point of view, since the US has the largest economy in the world, and the highest per capita income of any large country[0]. Where does the GDP is overstated line apply?

[0] 10 million or more people, by my current arbitrary definition of "large".