When you say:
"we cut regulations, which ends up hurting people and which companies take advantage of"

Think of this:
Companies take advantage when no other company is around to out compete them. This is why it's so important not to erect high barriers to entry around a market.

Instead, we have a lack of competition in health care in detail, throughout the whole industry. The reasons have to do with history. Its more important to see the general principles first.

Principle:
If you would like to regulate a market to "fix" something about it, there will be down stream ramifications that will later necessitate the regulating of something else - as various entities react to the original regulation so as to maximize their own well being.

Case in point: price and wage controls in WW2 induced companies to start adding incentives instead of salary raises for workers. Which incentives? Health insurance was a big one. Companies needed to step up production. The government errected wage controls "for the war effort" - so companies enticed employees to work more via other means.

Immediate consequence: Separation of payer and receiver of healthcare services. (I am assuming this is clear enough - the insurance pays for the service, the sick person receives the service, the doctor or other provider provides the service. Now there are 3 actors where before there were 2)

Long term effect:
Healthcare is expensive because the person receiving the service and the entity paying for it have been separated since the widespread adoption of health insurance "for everything" in the later 20th century. This distorts the perceived costs, allowing health care provider to increase costs without driving down demand directly. Instead, the insurance company is on the hook, and "I" can go to the doctor with greater frequency without commensurate increase in my expenditures. Cue the expansion of the healthcare industry, and a general rise in the _price_ of health care over time.

Long_term_response: health insurance companies hedge their bets and get into bed with "big medicine" and "big pharma". The health care industrial complex is born.

Longer_term_response: Long term rising costs finally induce broad measures to reduce the quality of service and make it more difficult to obtain service. Doctors are exchanged for nurse practitioners. Large companies make their health plans less and less attractive every year. The health care market enters a malaise.*

Summary thesis:
Without clear price signals, a market becomes more and more distorted over time.
People looking at "a snapshot" in time see a malfunctioning market, and ask for more regulation since "clearly, something is wrong". This tends to induce further distortions, and further calls for top down fixes. The few (the industry titans, and the doctors) benefit at the expense of the many in this system. To unwind this system, there is great risk of making things worse, as predatory situations will appear as the regulations are relaxed in an environment that has been specially adapted for them - people are vulnerable since the market is not functioning and a regulatory decrease is an exogenous step.

Importantly,
It's not that I, or free market people in general don't sympathize with people caught up in this situation. (We are caught up in it too! Most people are not tycoons) The (a) problem is that free market people sometimes call for immediate cuts in regulation - which are quite often unsafe. A general understanding of the situation is needed. -What is needed is a free market system, but a return to that system must be done in a smooth and safe manner. Unfortunately politicians have no incentive to pave the way to this in dollars. Instead, "the right" will cut regulation where it aids the medico-industrial complex, while quietly ensuring their asses are covered and their regulatory capture will reamin quasi-permanent, and the left will regulate to "fix that via a safety net for those on the bottom". In the end, the high and mighty will continue winning, since the tycoons are safe thanks to regulatory capture, especially with the masses placated with a safety net. And the malaise will only grow worse, and more confusing.

Side points:
-Some responses might be perceived as bad or good, but they are just responses - intelligent actors responding to a situation. In any economic situation, action and reaction, history and future - all must be considered in order to pass an intelligent value judgement. Hasty proclamations motivated by emotion are likely to miss important aspects of an economic trajectory - of economic dynamics.

-There is a duality in your argument, as in all things in economics. If you don't see this, then the politicians will have you.

-About the call for a "an actual quantifier of how much of a 'free market' something is": What kind of quantifier do you want? An objective measure? But surely you must know that any particular measure must be invariant to various kinds of hidden change in order to be objective - and nobody knows how to produce an economic measure that is really objective. This is why econometrics contains so much bickering over ad-hoc modeling and so many statistically based papers (across empirical fields which measure human subjects) have results that are hard to re-produce. I am sure some clever quantifiers have been invented, but when you do not even know the basic interplay of the dynamics, over-focus on some number -even if it is a good measure- is not helpful - since you cannot understand how good it really is unless you have the underlying processes well understood. If you argue that a different process is actually more important, then perhaps your "good number" may actually be bad.