Have you considered smaller insurers?
Smaller insurers have a smaller risk pool by nature. Therefore, they must either be very choosy about who they accept (which is problematic for people with, say, pre-existing conditions) or charge a lot of money for their premiums because they lack the leverage to negotiate with providers and don't have a large pool of insured individuals to spread costs among. While they might be a good option for some people, they can't (and won't) insure everyone.
Free competition is the key,
The problem is that health insurance (and healthcare in general) doesn't operate on free market principles. You can't shop around for the best price or highest-rated hospital when you're having a heart attack or you suddenly go into labor or you run over your foot with a lawnmower. On the insurance side of things, as I said, smaller companies have to cut coverage or limit access in order to compete on price with the giant conglomerates. While that may look like a "free market" at a glance, offering lesser coverage (e.g. a catastrophic coverage policy rather than full coverage) or blocking the guy who has a family history of colon cancer is not something that we should be aspiring to.
Realistically, if you have the principle that everyone should receive medical care if and when they need it, the only thing that makes sense is a single-payer, government-run system.
Blacks Lives Matter, for example, can offer their members health care and share the cost between members. They can cap the salary of a CEO at $100,000 and put it in their bylaws that every employee must make at least half of the salary of the CEO in order to keep costs affordable for patients. Anyone can incorporate and offer those same salaries and go into health insurance only, with a goal of making health insurance affordable for everyone.
What's the incentive for that? The government has a good reason to want a healthy population, but a private corporation? Please. Moreover, it's far from simple to run such a company.
A person should be able to CHOOSE insurance from companies other than the ones Obama and Pelosi decided to favor.
The ACA does not restrict what companies are permitted to participate in the marketplace. They must offer plans that meet certain standards (e.g. no rejecting people based on pre-existing conditions, no lifetime benefit caps, etc), but there's nothing that says that Joe & Mary's Family Insurance Co. can't list their plans on the marketplace. The main reason why you only see plans from the "big, fat, rich insurance companies" on the marketplace is because they have a huge risk pool, and thus can offer better rates for plans that meet the ACA standard. And people buying their plans on the marketplace are going to look at two seemingly identical plans and pick the one that costs $500 less per month.