That statistic is not terribly revealing, as I suspect that the vast majority of those 71% are relatively healthy/young people. I'd like to know the percentage of people who have had significant health events in the past 12 months that are happy with their coverage.
Using your insurance for regular doctor appointments or the occasional flu or bronchitis is quite different from realizing you have a serious chronic condition, or require a major surgery. You don't buy insurance for regular needs, you buy it for the unforeseen serious ones. These are rare, but it's here where the quality of insurance really matters.
Judging the quality of insurance from the standpoint of the "average person" is not the correct perspective. You need to judge it by the person who needs to file serious claims. From this perspective, 70% of "average people" being happy with their insurance seems remarkably low.
Government insurance, at least for federal employees, is actually very good, and for the same reasons you stated. There are few with as much bargaining power as governments.
But I don't quite get your point. You said that 70% of people are happy with their insurance. And my point was that doesn't seem very high at all as most people aren't that sick. You can't use a majority's viewpoint to judge the quality of a service that they don't use.