You're right, you can't throw money at a problem and expect results. However, you can attract brighter educators with more money. If someone is very good and has people interested in him, he can take X dollar from school system MoJo, or take X + Y dollars from school system HoJo. Hmm. Doing the same work, but getting paid more somwhere else. No brainer.
And if the money is going towards facilities. Fixing up buildings, adding computers, adding teachers with specific expertise, etc, then yes, adding more money can help the problem.
Of course, if the budget increases and all you get is admin pay raises, then there is a problem.