Of course that is exactly what your employer paid on your behalf before. So you're basically complaining that other people will now no longer be subsidizing your medical costs. A private system won't help you much. A public one will but merely because people who pay taxes will now be subsidizing you.
To see why this matters, imagine embarking on your "conquer the world" startup. You've saved up enough money for 12 months of runway. In a public system you are making 0 dollars and thus won't be taxed. However you will still have full health care. In a private system you will need to pair a fairly substantial amount (but probably not too much more than the amount of taxes that would go to healthcare in a public system) to receive coverage. You now have the following consquences: a) less runway or, b) more time saving up to get an equivalent amount of runway or, c) not having coverage and praying nothing goes wrong d) you just don't pursue your idea because it would be too hard.
Then, in a public system, when your startup succeeds and you make 10 million, you are taxed progressively on that money so as to fund the next hungry young graduate with a dream and a compiler.
I think it is pretty clear that the public system allows for more risk-taking. It is also important to note from your own example that private insurance is pretty expensive unless you are young, single, and without kids. This eliminates a huge swath of the population, all of whom might have different and awesome ideas. We bemoan the lack of gender and racial diversity in the computer field, but I think that the lack of age, lifestyle diversity in start-ups is just as important an issue.
That is the definition of insurance.