Analogous programs
This is essentially how the the insurance policies offered on Obamacare (a.k.a. the Affordable Care Act) insurance exchanges for people seeking non-employer plans work. In Colorado, for example, this exchange is called "Connect For Health Colorado."
It isn't exactly the same, because the group is everyone in the state who is not covered by an employer plan or with government health care, rather than literally everyone in the state.
Potential issues
There are some regulatory issues with this proposal:
There would probably have to be at least some exclusions, for people who receive health insurance through a private sector Medicaid or Medicare or Veteran's Administration plan or active duty military personnel coverage, or through a national or international health insurance provider for a multi-state or multi-national employer.
It might require an Affordable Care Act waiver from the appropriate officials in the federal government.
It could conceivably require a waiver of ERISA requirements (which governs employee fringe benefits).
It might have to be phased in to deal with existing three to five year obligations under union-management collective bargaining agreements.
But it could probably be done without all that much difficulty if a state decided to pursue that option.
The government would be a broker not a health insurer
Also, in these arrangement the state or local government isn't acting as a health insurer. Indeed, some states like Colorado, have state constitutional provisions that prohibit government from competing with private businesses in most instances. Most states have insurance regulations and government finance rules that would be incompatible with being a health insurer (e.g. many governments are not allowed to accumulate large cash reserves, while health insurance companies must do so). But what is proposed is really acting as a monopoly health insurance broker, rather than acting as a group health insurer.
Tax Issues
Another important issue would be how this is handled for tax purposes. The tax treatment of self-employed health insurance or non-employment related health insurance is different from the tax treatment of health insurance provided as an employee benefit by an employer.
When an employer buys a group health insurance policy for the employer's employees this is a deductible business expense of the employer that is excluded from the income of the employee entirely and never shows up in any way on the employee's tax returns.
If the employee chooses a plan individually on the exchange, this income tax and FICA tax exclusion for the employee would not apply. Something like this might have to be done through some sort of medical savings account or flexible spending account set up by an employer to gain some tax preference with employees who aren't self-employed. But it isn't clear to me that the same optimal tax treatment available for an employer provided plan would be available under the existing Internal Revenue Code for this kind of arrangement.
