I think the problem is you're talking about the ideal and not reality of many schools.
> 1) To push them further than parents think they should be pushed,

No Child Left Behind means the entire curriculum is slowed down so no child is left behind. If any kid excels or are just average, they're have to wait up because the numbers game keeps teachers from failing students.

>and 2) to socialize them. (Which, really, is just part of 1.)

I have met many homeschooled kids, and many public schooled kids. Guess which ones I see have the worse issues with socialization? Don't know how to interact politely with others, or have baggage that makes them afraid in uncomfortable social situations? Also turns out, kids have issues with socialization, shyness, etc regardless of environment.

>Ignoring all of the advantages that schools have -- mainly, more experience than makes sense -- even "bad" teachers are almost guaranteed to be better than a given parent. Because of their experience. They will have seen more kids than I can comprehend.

Teaching is not usually the career choice of the most gifted and talented, maybe in NYC or SF, but not most places, those that are burn out quickly as they're frustrated with the limits placed upon them by the system, the constantly being told how to do their job by parents, administrators, school board members etc who have no idea what is going on in their classroom. That and with a class load of 30+ students, even the most gifted teacher's attentions are divided to where they have to teach to the lowest common denominator, meaning those on the fringes of being high achieving or struggling are left wanting for attention, or worse the teacher gives those students a greater share of her attention, to the detriment of the rest of the class. The whole thing is screwed up and designed for teaching factory worker's children, not how multiple experiments have proved its better. Not to mention the push/pressure put on younger children, who learn better by playing anyway, see Scandanavia, they don't even put their children in school until they're six, but we're trying to teach math to 3-4 year olds whose brains aren't developed enough to grasp the fundamental concepts of quantity. Furthermore, a parent doesn't have to be an expert in all children, just their own. Even teachers are better with certain types of kids, this idea they know how to deal with all is silly, those kids that are different usually just struggle. I begin to wonder how much time you've spent in an actual school? Ever seen children be put in to a BEH program with kids who've been to juvie because they had a learning disability but were otherwise bright and gifted when put through tests? I have, it happened to me. We had to move to another county to fix the problem. You can teach to the individual needs not the whole class, so you only need to know what that kid needs, not every kid, again, its case by case thing, some kids are better off in public school, not every parent is suited to do home school or even able to. But two college educated parents can give most public school teacher's a run for their money on 1-3 kid(s) vs a teacher having to teach 30.

Peer relationship thing is again, a silly misconception, the peers I built relationships with weren't even in the same classes as me. I had a few friends from school, sure, but I also had friends who were cousins, lived in the neighborhood, friends of other friends, kids I met in Karate class, or kids of my parent's friends, etc. You know, like how adults make friends.

You assume too much isolation, you have little awareness of the secular education options out there for homeschool kids, the "hack your education" talk at Ted is a good example, a home school kid has more opportunities for education because the whole group doesn't have to be included, to say nothing of homeschool groups, where a CS major who does programming courses for his students does special sessions with whole groups of home school kids, etc.

Public school isn't awful, but it's a government program, and many times its not hard to beat the bare minimum that government program's provide. Should this be fixed? Sure, but the solution isn't to take away options from parents and kids who need a good education today, while the government figures out how to fix that.