That’s a societal decision we have to make. Should all children have access to good education - i.e. should we invest in public schooling? Or should it be dependent on your parents ability to pay in which case lets go all in on private.
Unlike many on the left, who despise monopolies in business and ruthlessly hunt them down, yet worship monopolies the government has (like in public education), I think public dollars should be allocated to the parents to make decisions they feel are best for their children. Saying “just move if you don’t like your public school” betrays the reality of how difficult it is to relocate for the non-tech elite who can’t just work from anywhere.
I don't know how to fairly evaluate a school's performance. I guess we could ask if the students perform better in the job market 10-20 years later, but that's not obviously not helpful or useful information on which to make a decision now.
I get why the voucher system is appealing on first glance. After all, you often can evaluate the quality of goods and services. You can, say, estimate that a shirt you find at the mall will last a few years or that it will fall apart in a month. I just don't think you can do the same for schools. I have a feeling most people will evaluate education based on their beliefs (religious, etc) and the grades their kids "earn".
Further, how many Harvard-level grade schools do you think we can run as a society? Given that private school pay is substantially worse than public school pay on average, how could one say that private schools, again on average, would outperform public schools?