The tricky part is that the current system may act as a filter where the best people will still try to hide their methods and get into the existing Journals because it's best for their career - so you'll get lower quality requests for funding.
I don't understand why Universities don't work together to push towards open access, but I guess they're interested in Journal prestige too.
As a side note there's also the issue with papers hiding critical components of their research so they can't be reproduced (so they can make companies later) - this is really not in the spirit of science.
Yes you could be right about that. It's possible that in the short term an individual funder could see a drop in the quality of applicants -- but I think this will be offset by massive increase in overall impact. I think science funders underestimate how much power they have to compel change.