Yet golden rice was made mostly through state funding - patents aren't the only way.

One of the slightly unfortunate things about golden rice (and there are quite a lot of them) is that the state funding it was developed with required them to patent it and sell the patent to the private sector. Though not patenting it would've been of limited help due to the thicket of patents covering the technologies they used.

Why were they required to patent it and sell the patent?

I think the organization funding it (which if I recall correctly was the EU) wanted the university research they were funding to turn into commercial products, and so set a blanket rule requring patenting and attempts at commercial licensing. This is a fairly common view on government-funded research these days unfortunately. There was a published report by the main researcher somewhere with more details.

Well, that was a stupid idea, then. Instead of selling the patents of public research to private monopolies, they should have made them available to everybody.

There is nothing stopping them from doing the same for other uses. Yet, they don’t.

I wonder if something like the Kickstarter model could be applied to this sort of research.

Novel investment models and the increased accessibility of bioengineering technologies to individuals have the potential to revolutionize the field in the future.

Witness the blossoming fields of biohacking and citizen science.

The fruits of individual, independent experimentation and development may get out in to the wild, if they haven't already, and no patent is likely to stop them.