If it's an individual health insurance plan, good luck getting them to pay a major claim. Just because they are required to, doesn't mean that they will do it without putting up a big fight. Fortunately the Affordable Care Act eliminated some of the worst abuses of the health insurance industry (e.g. kicking people off plans once they got sick, denying coverage because they failed to report an old irrelevant preexisting condition, etc). But you are deluding yourself if you think that mere legal obligation will stand in the way of their profits.
Furthermore, as important as the ACA was, there is still a lot of abuse that remains unregulated in private health insurance plans. For example, it is still legal (at least for a couple more years) for insurers to bury exclusions for specific conditions deep in the fine print, which no one without an actuarial or medical background would know if they need or not.
The people do.The government regulates that. If the government is the one you hold the contract with, who regulates them?
Furthermore, your argument doesn't make sense. If you trust the government to regulate private insurers effectively, why would you not trust the government to make policies for itself effectively? In this logic, all that private insurance does is add one more layer of bureaucracy.