Most people can only get group health through their employer and many employers either don't offer it or make employees pay for it out of pocket. A large segment of the population simply has no way to access a group plan.
You have actually made an excellent argument for single payer health care. A private health insurance system does not work because people have no choice whether to get health care or not. People have a choice whether they want to drive (auto), own a home (homeowner's), run a business (CGL, property, E&O, D&O, etc.) and so on. You can always live in an apartment, take the bus to work and walk to the store. But, if you have diabetes, you have no choice about whether you can get treatment for it. If you don't, you die. In the current system, people who are excluded for preexisting conditions get care, but the insurance company doesn't pay. The providers then overcharge other patients, particularly the uninsured, to compensate for the non-payment.
In a single payer system, everyone pays through taxes and costs are kept down because there are no profits on the insurer side, administrative costs are reduced to a fraction of what goes on now because you have a single system for payment instead of dozens, and the government run health insurance plan has real bargaining power with providers. But the real benefit is that no one is ever put into financial jeopardy by illness. The majority of bankruptcies in the US are cause by medical bills. For each bankruptcy, there are a myriad of losses down the line to the creditors. Under single payer, these would all end. Also US businesses would be able to compete with the rest of the world. Japan, Canada, Germany, France are all emerging from recession because of rising exports. The US is still stuck in recession because we have lost a massive chunk of our export industry due to the high cost of health care that employers are expected to provide in the US.