See https://en.wikipedia.org/wiki/Public%E2%80%93private_partner...
> healthcare
The large healthcare providers are already run like private businesses, because they are private businesses.
> education
Please no.
This is a favorite pet issue of the rich and powerful. Their ranks are littered with failed school reform activists. Sometimes those folks even have the self-awareness to realize they failed, e.g., the Gates Foundation. More often, they keep on proposing the same old ideas that are continuously tried and fail, while sending their own kinds to expensive private schools or public schools in the priciest zip codes in the country. Schools that are, aside from their wealth, operated in a quite traditional fashion.
If you look at education outcomes by country, there's really only two things that stand out as invariants that separate education systems that work from those that don't:
1. degree of heterogeneity in schooling (school choice without some strong equalizing effect on quality just rearranges the deck chairs on the titanic), and
2. level of respect for educators (pay != respect).
The "rich people can fix this" attitude is truly both a symptom and a cause of the biggest issue with America's education system: teaching is a low-status job (even places where it's well-paid).
When it comes to education, Americans respect literally anyone's opinion more than the opinion of educational professionals. Hell, even random people who were literally born into wealth get more voice than the professionals who spend their lives in the classroom. If that doesn't change, nothing else will.
Online Education? Coursera, Lynda, Pluralsight, Kahn Academy, Youtube for crying out loud.
Healthcare? - Healthtech is becoming a big thing, I had the opportunity to interview at a company that is currently working on a private health cloud where they aggregate all of the info about you, your smart watch, your phone, etc. (Note: That you choose and opt into) to make it available to determine the best outcomes for you personally and help the Dr. working with you practice individualized medicine.
You list other things, but hospitals are being changed by technology at an amazing pace, the fact that now during a pandemic I don't have to leave my home to see my PCP, you think that was the burdened bloated Hospital beracuracy that came up with that?
You then list roads and bridges, but the point is that yes, VCs and entrepenuers could figure out how to solve those too given time and a enough freedom, even we can't see it right now, it may be because our vision is just too limited to see what could be, that's what makes entrepreneurs, real entrepreneurs, different they are visionary.
Brightline / Virgin Trains USA [1]
Texas Central Railway[2] which aims to introduce Shinkansen-class high speed rail connecting Dallas and Houston
Also a healthcare app won't replace those things but robots, computer vision, and ML could. Or they could make it so that a Dr. instead of doing 8 surgeries a day could do 80, or a Radiation machine that is totally self managed. You pop in it runs the test then it can interpret results with a 99.99% accuracy. There are lots of ways tech can augment and improve healthcare.