Its not just the lower ratio of teacher to student that private schools can provide, but also the resources of equipment, etc, public schools struggle to keep the lights and heat on, private schools have pools and full on audio-visual labs, and whatever. But it depends on the expectations of what the child is going to accomplish. Are they able to also fully fund university tuition and expenses, and is the child progressing well academically (or even, as mentioned, music and arts) If not, what is the preparation supposed to be good for, if they end up not being able to move on to the next logical step.
There is more to it than just coming up with the school tuition, and even wearing uniforms isn't going to even things out. Many of the students are going to be from wealthy families, it will be the off-campus activities that will set the scholarship student from the others. If you are taking the kid on winter vacations in Aspen, and spring break in Los Cabos, then sure. It may make a difference, it might not. It isn't specific to private schools, tho, cuz public schools can still show up the class and income differences.