Yes, if wishes were horses everyone would have a Merry Christmas.
So, let me break down why that doesn't work.
First, great anecdote that you taught your nephew. Irrelevant though it was, it's good to know that you think you helped. And it's good to know that you understand that it is something that is not financially possible or practical for all parents to do.
There is also something called the GED, the SAT, and the ACT; these are required to at least achieve a high school degree in this country depending on your choice level of education or state; even if you successfully weed out any curriculum you deem to be "agenda-driven" as if it's counter-intuitive to the standard tests; they will still have to take those standardized tests to get their degrees. And while, yes, many home school programs will facilitate these basic levels; they're still, again to be unlikely to advance their chances of getting a job that pays more than minimum wage.
Do you know one of the reasons why the left are constantly advocating for raising the minimum wage for entry level jobs? Because many of those jobs are only for those with at least a high school application or less. The only way to get raises or advanced is with a college education.
Even if you successfully keep your child away from "draconic leftist teachings of teachers" and "leftist colleges" they're still going to be snatched up by the jaws of the unemployed and easily coerced into thinking ANYTHING is better than what they have now. (Like communism.)
Yes, parents absolutely should be involved with their children's education but I'm saying they should not com it. It's like saying...
"You know what the problem is with corporations? They're not completely controlled by average middle-class working Americans! We should let middle-class working Americans run corporations!"
And hey, that can be described as socialist thinking too! (Workers Unions)
Even if you're going to try to tackle this from a commercial perspective, as if that's any more viable, that when you say parents should have "complete control" over their child's education, you mean they should have "viable choices" on what is taught, that is still not going to address any issues you currently have with the current way schools work. Because, arguably, we already have that with home school programs, charter, public, and private schools.
So, I guess what do you mean parents should have "complete control" when they already, technically, do?