This can **not** be fixed because it is impossible to have it both ways ( as you so wisely pointed out). The government schools can not be both godless ( which is **not** religiously neutral in content or consequences) or God-centered ( which is also **not** religiously neutral in content or consequences).
It is impossible for any school to be religiously neutral because such a state of philosophic neutrality is impossible in the mind of any sentient human.
The NON-neutral religious worldview now taught in the government K-12 schools is godless secularism. The children in these classrooms will learn to think and reason godlessly. They must just to cooperate within the godless classroom, read their godless textbooks, and do their godless homework.
This is only one reason among many why government schooling should be abolished. They are are First Amendment and freedom of conscience abomination.
I don't think you have thought this through. Most people would receive a very poor education if an attempt was made to universally educate in the home or in private schools simply because most parents work and most parents cannot handle the wide range of subject material. As I said before, private schools were always too expensive for my family. Our denomination is too weak in the South to be able to support any sort of school. I could not home school because I worked for a living.
Now, let me tell you what I am teaching after hours in high school. First, I teach metalworking including hand tools and machining. The county just bought me a metalworking shop including a nice lathe and mill plus scads of tooling and precision measuring tools. That was a great deal of money. Can you do that in your home? I will teach the kids to fashion parts made of metal. Next I will teach courses in mechanical movements. They will learn to make grippers, positioning mechanisms, and so on. I will teach them to choose motors and gear boxes, or to make gears of their own design. I will teach a course where students will learn to model mechanical components is 3D, and to use those models in assemblies to check functional behavior before the design is committed to metal. There will also be course work in digital and analog electronics, including the use of micro-controllers for control of mechanisms. There's more! I will teach them to program in C/C++, and from there students will learn to make the machinery move to accomplish predetermined tasks.
Where in there do we have time to cover everyone's religious dogma. We don't. I won't learn that stuff so that I could teach it anyway. Do you not yet understand? Now, factor in that there is a great deal of secular material to learn in order to function into today's society. A small group of teachers do not have the knowledge to cover all of the subjects to proper depth. There is not sufficient time to deal with religious topics if the necessary secular topics are to be taught properly. It is certainly not practical to have religious studies for every denomination of Christians (lots of 'em), Jews, Muslims, Buddhists, Shintoists, etc in order to preserve everyone's religious preferences. It is far better to leave all of that to the individual families and churches. Anyone who wants to mix education with religion is delusional. I live in the Atlanta area. All races are represented in school in large numbers. So would their religious entanglements if that were allowed. There would be no time to teach the basic skills.
You are correct when you assess that some things are done in schools that qualify as indoctrination or prohibition of religion. THAT is what should be combated, not the concept of public schools. Schools are not religious neutral because conservatives do not engage, they duck and run. It is imperative that conservative people become confrontational at every level in order to enforce Constitutional neutrality in governance.
Conservatives seem to have no skill in "political tactics". They bitch and blog, and then go home to be schooled. They aught to stand their ground. Schools were infiltrated by leftists according to a plan. I personally am infiltrating schools according to a plan. If conservatives will just look at my measure of success and follow suit, a proper balance could be restored. But NO! All you wimps want to do is bitch and duck and belittle the headway I am making by taking matters into my own hands. You people think voting does something! It doesn't! Only action does something. The leftists have known that for decades. Conservatives NEVER catch on. No wonder they think conservatives are stupid.