A problem with the public sector in this instance is it has money to spend, but no way of allocating it particularly well.
It will just pick the best allocation metric it has available, even if that metric would never stand up to scrutiny in the private sector, or any more directly measured domain, public or private.
I think the state could simply allocate money to long-lived scientific institutions and let the experts there handle things as long as there is no obvious corruption.
Self-regulation has a tendency to either work well for a few years, then gradually become corrupted... or be corrupt from the beginning.
A distressingly high percentage of humans like zero-sum status games. More people are happier when status is recognized as a semi-unbounded positive-sum game.
It will just pick the best allocation metric it has available, even if that metric would never stand up to scrutiny in the private sector, or any more directly measured domain, public or private.