Normally that would be true, but under ACA profits are capped at 25% of premiums. They can increase profits only by increasing premiums, which can be justified by rising costs. Private insurance companies in America have no incentive to lower healthcare costs.
Well, that's true if they hit the cap and have no competition. If they don't have a 25% profit margin yet, they can still try to increase profits by cutting costs. There is also the strategy of lowering prices to increase market share.
The downside I was hinting at is that as a patient, cutting costs isn't always what you want. If it means more prevention or paying less, it's good. Otherwise it'll probably mean worse service.