First, US insurance companies are often pseudo non-profits, with profit caps. [1]

Second, another model is Medicare Advantage where beneficiaries can choose a private entity to administer their Medicare plan.

Third, most commercial insurance in the US is not actually underwritten by insurance co's, but by "self-funded" employers. In this case, the insurance company is merely a third-party administrator (TPA) for the employee plans, using XYZ's bank account (and trying to save them money, offer health incentive programs, etc., to secure repeat business). Most companies with more than, say, 500 employees operate this way, though usually you won't even know this is the case.

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Finally, if by

> do we want to provide universal access to healthcare?

you mean "do we want everyone to have healthcare?", the answer is an easy "yes". We want universal access to housing, healthcare, employment, etc.

But if the question is "do we want the government to use taxes to ensure that everyone has healthcare?", the answer is far more divided. There's a lot of interesting examples elsewhere. But as you've mentioned, the US government has already dipped its toe into those waters with the VA, and I don't think it's been perceived as a success.

[1] http://thehealthcareblog.com/blog/2012/02/04/does-obamacare-...