I would argue the lack of transparency in the marketplace and the inability for anyone to make an informed choice of hospitals or what it will cost them (the linked article seems to indicate the consumer has a choice to go to a "nice" hospital or a "not so nice" hospital, but most times we don't have the choice when we need care - it's close to our home. The times we do have the choice (sometimes for birth, cancer treatment, etc) is where you see hospitals investing heavily to get high margin services from consumers.
Does Employer-provided care make medical care unfair? One could argue that employer-provided care with the tax incentive might lead to better resources for an employed person electing that care, but any time money is involved you will likely get better care. EDIT: More succinctly - There will always be some inequality / unfairness in care because there will always be inequality of wealth. I'm not sure explicitly here that employer-provided is the problem or that any subsidized care will create this kind of inequality.
I'm not sure you can argue effectively that granting the same medical care to all as a matter of law will fly because people will always find a way to get better care if they can afford it. Does employer-provided do this at a larger scale? Maybe?
The best way to ensure fairness and efficiency is to get the data out there and available, and make healthcare more like a market instead of a black box.