Public sector at least has the pretense of trying to do what's best for society.

The market is a race to the bottom; public entities have their goals stated expressly, and there will always be some idealists who genuinely care about living up to those. Either way it comes down to people just deciding to do the right thing; there's no way around that. But capitalism gives otherwise-good people the excuse to do the wrong thing without feeling bad about it, because it's "just business".

When reasoning about performance, an institution's mission should treated like a student's name. The information is irrelevant, but even for an honest and conscientious reader, the cognitive traps laid by its availability are too dangerous.

Incentives are a decent guide.

The private sector has money as a motive.

The public sector has motives as well, like money (continued funding), political pressure, etc.

You’re just exchanging one motive for another and not really solving the problem.