One of the latest papers by Hanushek, the person who tends to be cited by those against public school spending "U.S. SCHOOL FINANCE: RESOURCES AND OUTCOMES" gives a more mixed overview. Basically saying it matters somewhat depending on what it's spend on and only moderate improvements.

The paper gives an overview of more recent research mostly using quasi-experiments. Before 2000 or even 2010 just doing some linear regressions was more common. Anyway my view is similar to Hanushek in short/medium terms. I do believe long term the pay of teachers, and therefore extra spending, is an important factor in keeping teaching prestigious compared to other jobs. In the US partly because of its strong private sector this is a lot more difficult/failed, I'm not sure if it's possible to fix since spending is only one part of it.