No they do not. They have a definite hand in it ($$), but Teachers unions and public opinion at the school board level is what runs education.
And I (think I) know you are exaggerating here, but it is not even close to the truth.
The education received in a public school is not exactly, but it is close to, the socio-economic base of the district. Wealthier areas have more money for the schools to hire veteran teachers and ample supplies. The parents are also usually higher educated and education is a priority. The parents are also more likely to be home in the evenings (rather than the second job)
In areas with a high % of English learners (also generally poorer districts) your statement about the 3rd grader in private school could at least be correct when compared to that of another 3rd grader. The amount of time spent teaching children English, only to have them go home to a Spanish only household is detrimental to the learning of native speaking children, period.
However, not all private schools are the same. Most are church based. These schools always have music, and usually a religion class.P.E. is often left out, but the amount of the 3 r's suffers for the hour of religion. They do not teach spelling or fractions on Sunday, so why during the academic week. These children often transition to public school at the middle and High school level as the expense and pressure from the children to be "normal". Not always, (some schools are excellent...and expensive! avg. pay is $10,000/yr less than public school!) but often these kids show up to public school behind in math, not to mention ruined for science!