Re: so, let me get this right...
Not quite. All "social welfare" is in that 30%. Health care is about half of that (18% of GDP in total, from memory).
So, no, it's not all about the health care system.
And tricky dicky? Nixon? Not all that much to do with it as far as I recall. The idea of health care insurance being provided by the employer started in WWII, under FDR. Wage constraints (hey, there was a war on!) meant that employers competed by offering better health care insurance. It's also true that health care insurance is tax free: it's an expense to the company, so part of costs, not taxed. But it's tax free to the recipient. If you were paid the money and bought insurance with it you'd pay income tax on that money. But if the company buys you insurance then the employee doesn't pay tax.
Changing that alone would make the system rather better.