The Obamacare tax was something like 3%, and only for the extremely wealthy.
In contrast, 33% of US healthcare costs go to insurance overhead, and >17% of our GDP goes to health care, so nearly 6% of the gdp goes to private insurance middle men.
Source on 17%: https://www.statista.com/statistics/184968/us-health-expendi...
Now, the entire federal government is currently spending 37% of GDP (though that includes deficit spending), so you posit that at least ~ 1/6th (6/37) of government spending is some healthcare program.
Source on 37%: https://tradingeconomics.com/united-states/government-spendi...
Medicare and health costs ~ $1T of the government’s $3.8T budget, which is a bit over 1/4th of the federal budget. That does meet the threshold of your claim, but only if much less than half of Medicare spending goes to actual health care.
Source on 1T Medicare, etc spending: https://www.nationalpriorities.org/budget-basics/federal-bud...
This study from 2001 concluded Medicare is about 80% efficient (20% waste): https://www.nber.org/papers/w8395
Note that the 20% Medicare waste number from the NBER study is not apples to apples with the 33% administrative overhead number, since it includes ineffective medical care (and all other wasted spending) that did not improve patient outcomes in the 20%, but the 33% number for insurers only counts useless paper pushing, and not ineffective care, etc, etc.
So, private insurers really are squandering many more healthcare dollars, and are also much less efficient than the federal government.