No, the millenials will be paying for the older generation's retirement and healthcare via Social Security and Medicare.
My guess is younger generations, once better politically ensconced, will flip the script on boomers by funding broad social benefits out of social security and Medicare. (For example, by reducing Medicare benefits while making them available to all age groups.)
The boomers will be a unified block all voting for social security and health care for the elderly.
Everyone else will be split around various special interests.
So maybe in 20 years?
I think you’re overestimating the probability of this factor. Twenty years isn’t a bad estimate, but I’d guess closer to 10.
(Not that I actually believe there should be a privatised healthcare system.)