In an ideal world (according to economists), this isn't an issue. In the world we happen to live in, basic health research is publicly funded. Maybe it shouldn't be, and research funding should be purely commercial, or philanthropic. But with public funding crowding out commercial and philanthropic funding, banning public funding for some kinds of research has a similar impact to banning it.
How can public funding 'crowd out' other funding? There are more 'fund takers' than there is funding available, so if there were private funding available, it would be put to use. I don't understand your reasoning.
Research funding requires scientists. There's a relatively fixed supply of people who want to do research. Those who can get government or university jobs. Those who can't get jobs in the private sector. If you can't find any scientists, you don't employ any in your research lab.
This is an empirical observation, and isn't based on micro-economics, though you could frame it as such if you wanted. It's not true in all cases - Google might lure a few top computer scientists away from academia by giving them massive amounts of funding. See, academia has non-monetary rewards - the glory of working for as an academic (this may be a con), and government jobs look stable and for the public good (though libertarians might argue). Thus public funding secures the scares resource (talented scientists) at a lower cost, and private funding goes down (supply and demand, unless research happens to be a Giffen good, which I doubt).
