Good news, the Royal Society tries open access by Stephen Pincock:
It seems to me most grants for scientific research should require open publication. I can imagine exceptions, but it seems to me that the expectation should be for open publication, in this day and age, and only allow non-open publication with a good reason.
For public funded research this open access expectation seems obvious. For private foundations in most cases I would think open access publication makes sense also. What business model is used to allow open access is not important, in my opinion. The important factor is open access, how that is accomplished is something that can be experimented with.
I would see nothing wrong with additional publication in non-open access journals so long as publication in open access manner was significant. How privately funded research would be shared should be up to those doing the funding. If I were making the decision for a university I would have expectations that we publish openly. However, if a private university chose not to do so (for research they fund with their own resources), I think that would be their choice. Granted such clear distinctions are not common in the real world but this can give a sense of what seems reasonable to me.

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