That is part of the failure, and I am deeply affected by it right now. I am a pretty good teacher, and I watch terrible teachers get paid more than me because they've been at it longer. I can't pay off my family's student loans, and I can't afford anything more than a small condo.
But I still don't think privatizing education is the answer. There is always the possibility of taking education back from the politicians, and setting up a system that does incentivize good teaching. It's not as simple as paying teachers more if their students pass tests.
One fix that would go a long way is restructuring our approach to tuition in service sectors. If you take away my student loans, I would be a happy, hard working teacher the rest of my life. I will get some portion of my loans forgiven for teaching in a high-need area, but that won't go a long way. The same goes for other service sectors, where a reasonable job will leave you paying off student loans until you are past retirement age.
There are bureaucratic fixes. You can give more professional freedoms to highly-effective teachers. Measuring effective teaching is difficult, but not impossible.
As soon as you give up on public education and only see privatization as the answer, you give up on addressing the education gap between different socioeconomic groups.