But if private industry profits less from the research, they will pay less for it. Pretty quickly you will see two exclusive sets appear: a little open research funded with a little public money, and lots of proprietary research funded with lots of private money.
I don't think it would be as complete as you imagine. There would be companies who would forego keeping results private to save the expense of building their own lab facilities and hiring people to run and work in them. That public resources can be used for private results is a back-door subsidy for commerce.