"Supplementation" meaning what?
Your very first example, state welfare, is in direct and obvious contradiction to the free markets capitalism prescribes.
"If we relied exclusively on capitalism to meet our needs, millions of people would die, because capitalism only cares about profits and the concentration of wealth"
Incorrect. Capitalism doesn't care about anything at all. Only a mind, an entity, can have preferences or priorities. Capitalism is a prescription, a set of ideas, nothing more. To claim that capitalists care only about profit or concentration of wealth is demonstrably false.
" *That* is demonstrably true,"
Of course it's not. For one thing, it's laughable to point at our current system and claim it's capitalist. Capitalism prescribes free markets, and there are thousand s upon thousands of government interferences in the market.