I think we should scrap Obamacare entirely, and what Congress should do instead is permit the free market to work by allowing health insurance to be purchased across state lines as automobile insurance is.  The point you make about companies not paying a claim based on an uncrossed "t" or an undotted "i", I've not heard of, but they could deny the claim, and perhaps congress could make law against that, a "health insurance good faith law" or something. I think the interstate trading of policies (rather than the intrastate system we have now) would do more to lower costs than anything GovCo could ever do. Also, some state laws require mandatory crap to be covered, such as mandatory coverage for "port wine stain birthmarks" or "male pattern baldness" and it is things like this that jack rates up exponentially.  I currently can find my cheapest policy for $241 per month (x 12 = $2,892 per year), or I can pay my bills myself which is (per year) 2 doctor visits (one with blood work), $185; Prescription pills for hypertension and high cholesterol, $518.72; grand total per year? $703.72, not counting an emergency. So, I save $2,188.28 per year by NOT having insurance.
Secondly, insurance is neither a right nor a privilege, it is a commodity. It was during World War II when GovCo instituted "Wage And Price Controls" that employers, who could no longer offer legally a job to the best candidate for the highest dollar amount per year, decided to sweeten the pot to entice the candidate by offering perquisites (not "perks") such as company-paid health insurance for his whole family, paid sick leave, and paid holiday leave. Today, the dumb masses believe that the employer is responsible for employee healthcare which led to popular support of Obamacare which passed the House by a mere 220 for and 215 against, and the Senate passed a different version which originated in the Senate which passed the House 219-212 for; but wait, according to the Constitution, ANY bill regarding confiscation of Citizen's money MUST originate in the House. This bill confiscates MY money by MANDATING that I purchase a product which I have heretofore demonstrated I do not need.  The "Commerce Clause" argument is out the window with the knowledge that the bill the POTUS signed was originated in the Senate, and I hope the SCOTUS recognizes that. And Kagan should recuse herself from this case as she was the Solicitor General under Obama WHILE the law was being written! 
On the whole, institutional "free" healthcare is a bad idea, never mind who might be implementing it.