I'm not convinced that's true---although I'm not sure how much my perception is reliable given that I've never been inside a president's office. That being said, at least the perception within universities is that presidents are often responsible for some very consequential decisions.

The most obvious example is internal budget allocation, which can make or break entire programs.

This article from Bloomberg has lots of good quotes in it, here's one:

"President Hatch's compensation over the course of his tenure reflects his exceptional leadership," trustee chair Donna Boswell said. The school has raised nearly $800 million under Hatch, she said.

Right, no question that fundraising is an important, perhaps the most important, part of the job. I'm just saying that making mission-critical decisions about the whole institution is also in there.