"Can I take that portion of my tax dollars and give it to an organization that I feel does a better job?"

I myself am not saying there is no place for government. Certainly I believe that the welfare of our children is to some degree the responsibility of all, thus I would not say that paying taxes for parts of government we may not agree with should be "optional".

I would also say that taxes to help educational institutions, fund security, courts, etc. is very necessary.

What gets me about "public school" and vouchers question is that we already DO THAT for kindergarten (tax deduction) and for higher education (the college fund 401k style tax free savings accounts).  

Why not have the same for secondary education?

Obviously - we could not change the system overnight (witness ObamaCare) so a good solution might be a voucher system that at least partially reimburses taxes paid by those whom are sending their kids to private schools and thus not stressing the public school system.