The answer depends on what is meant by Duality since it has a few applications as a term and concept.
What I meant: If the concept of duality is pretty much universal (as I think you claimed), what happens when that concept is applied on itself? This is a bit like someone saying "There is no absolute truth." and a second person responding "Is that true?"
The government shouldn't have motive to put competition out of business because their existence diminishes the required effort to fulfill their role. If they cannot provide enough value to be useful at a zero sum profit, then that's probably a program that won't need to exist. The governmental program would naturally fill a role of simply keeping businesses from gouging the market. Again, our mail service is a great example of this. A 'way' doesn't need to be found - it's just a naturally built in benefit of having both private and state service options in an industry. Balance is found easily in this sort of scenario so long as selfish machinations are not corrupting the scenario, which is always going to be a problem to try to protect a system against no matter what the system is.
As long as selfishness is part of being human, it will be there - on
both sides. And as good as "checks and balances" are, such a system ends up either with an ultimate checker who is not checked (in politics this is often the high court), or a "circle" with several institutions checking the next one until the circle is closed. The first can corrupt very quickly, unless you are lucky enough to have supreme judges that are completely beyond reproach

, and the second can also corrupt, although it should take a bit more time until every institution in that "circle" has been corrupted. Since the people cannot do the checking by themselves, there doesn't seem to be a third option.
There are two particularly strong reasons why I'm especially distrustful of the politicians in this: First, because they can directly wield the kind of power that private interest groups can "only" try to influence, and second, because other people seem to be much less aware of untrustworthy politicians compared to untrustworthy business leaders, even today. They way some people talk you would think that elected politicians were (by their very nature) completely incorruptible - I don't think I really need to address this here, anyone still believing that should take a good look at the current state of the "sphere" we are on.
If politicians aren't beholden to their voters throughout the times between elections, they aren't going to do well when elections come up. Profit cannot be the only motive in a system or the system will immediately begin to find ways to profit from all possible strategies and many of those do not concern the wellbeing of the consumer, the nation, the world. All means nothing if you do not profit to the fullest extent in a completely free market controlled system and that WILL eventually undermine and collapse the system. A profit-driven health industry cannot truly succeed in its role when the health of their clients is not profitable.
You vastly overestimate the memory of many voters, I think. And yes, making the actual health of the patients profitable would be a very good thing - like the doctors becoming some kind of "health insurance" themselves and you pay them as long as you are healthy (one of them, whom you picked to "guarantee" your health). And when you are sick, you notify this doctor and immediately stop your payments - this way you will get an appointment quickly

, and this doctor will give their best to restore your health quickly as well. You even remove any motivation to hold back effective medication, because "your" doctor is just as interested in you being healthy as you are yourself.
Find out what people should want, and turn that into their own best interest.
Slavery is pretty much the end result of an out of check free market policy, as history shows. When it begins to cost more to live than you can earn in your life, if you don't have an option to sell yourself into indentured servitude, you just end up dying an early and avoidable death. It's important to understand that we're headed in that direction for so many if we do not demand a system that keeps the right to live from becoming prohibitively costly. I realize it's been worse for a lot of folks in the past. But we're trying to find a way to have a system where all people can remain free here. And to do so we cannot allow for such drastic wealth imbalance. I don't deny it can get, and probably has in the past been, much worse. My comment of how it's never been this bad would need to be taken to mean in recent history.
Sorry, but history is something you really cannot argue from. There are so few examples of a true free market in history that it cannot serve as any kind of proof. And I certainly agree that extreme poverty needs to be dealt with. I just don't think that wealth (im)balance is a particularly good standard - I would much prefer checking the level of living of the poorest, say, 1 %. That way, you can call anything that improves the standard of living of the poorest (whoever that may be) a good thing, even if rich people happen to profit from that development as well. And (even more importantly) any development that ruins the poor even further can be called a bad development even if the rich suffer as well (like: bombing the entire country into the stone age - that certainly reduces the gap

).
The last example is also the reason why you should never want your paramount desire to be that people you hate are destroyed - the "best way" to achieve that might hurt many, many innocent people.
OTOH it is certainly correct that the gap has never been this wide ever since the end of the Second World War.
It could be necessary to reconsider a few points if automation really ends up removing most of the jobs, but I am a bit sceptical. It still remains to be seen if "strong AI" becomes possible one day, and there are strong reasons to believe it won't happen (
https://en.wikipedia.org/wiki/Rice's_theorem could be Computer Science's Second Law of Thermodynamics, with strong AI being like a perpetuum mobile). Without strong AI, jobs that require human reasoning won't vanish, and having a sufficient number of jobs could end up depending on the quality of education, making sure that the remaining (and by nature very demanding with respect to intelligence) jobs can be done by the people that are there.