I also believe in free education, which includes research facilities.

As opposed to now, where companies copyright and patent their medications and sell them ultimately to...taxpayers who pay them billions of dollars a year in just out of pocket costs, let alone the scheme that is the American private insurance/healthcare system.

If taxpayers had to fund drug companies or research institutions for R&D without the insane middleman that is the private healthcare/insurance system, it would cost a fraction of what it does now.

On top of that, this assumes that people won't do research for the good of society vs becoming filthy rich, which is a false assumption driven by Capitalist propaganda. Remember that the ultra wealthy CEOs and executives of these drug companies aren't the ones doing any of the actual work or research. That is all done by scientists and engineers, who make a decent living, but none of them are incredibly rich from it, classic Capitalist exploitation at work.

Often times these drug companies (and the private equity firms that own them) don't even primarily do R&D, they just purchase the patents and IP rights to drugs that are already on the market, and once they do that, they jack up the price often by hundreds of percent to increase their cashflow.

That cost gets sent to the insurance companies, which of course, they pass on to consumers, raising our healthcare prices year after year.

I want medical researchers, scientists, and engineers to make a good living, a very good living, their work literally saves and improves hundreds of millions of people's lives worldwide. But you don't need a CEO or executives, or private equity firms owning that space and making insane amounts of money, you just literally don't.

There are millions of very smart and passionate people around the world who want to do this kind of work because they enjoy it, and they want to make a difference. Providing an open and rigorous academic and scientific structure to study and practice this is all you need. That already exists today, many medical breakthroughs came from publicially funded research institutes, which is the way it should be.

If the entire world had access to free healthcare, chances are research and development would grind to a halt unless they also funded research and development. Taxpayers would need to be willing to pay a company hundreds of millions of dollars if they discovered a useful product.

I don't see why it would. A company would still invest in research if they thought they had a chance to sell it to the healthcare system, for example. It wouldn't be the first nor last time something like that happened, and the latter case isn't too different from how it works already.

Consider insulin, for example. Research into it and drugs for treatment of diabetes doesn't happen exclusively in the US.

…it can work in theory, but I’m not sure if it would work in a democracy.

And yet many democracies have UHC... and publicly-funded research.

Penicillin was invented in a public university lab. We used to have a strong public research in the health sector worldwide.
Then neo-liberalism managed to convince politic leaders to progressively shut down public research and subsidize more and more the private sector.

https://www.ineteconomics.org/perspectives/blog/us-tax-dollars-funded-every-new-pharmaceutical-in-the-last-decade

This is not unique to the US. Your taxes pay for R&D, then your insurance (or your wallet…) pays to get the drugs you subsidized. And the private sector does not need ginormous margins to stimulate private R&D spendings.

Social medicine only enhances research and development because the studies can be conducted in a wider manner.

R and D is funded in nations with universal health care all the time. There's always been a pipeline and interchange between public and private health research interests. I'm not sure what your point is.