Keep in mind, I was not a student in the 50-60s (I was in 1-12 in the 70s)
Here is my take on one of the reasons:
Women's liberation. Before I go further, I am not suggesting that there be some sort of de jure or de facto return to limiting the jobs women can have, but the expansion into other fields hurting teaching and the education system. My argument:
Back in the 50s, women had few career options (and many were just jobs to have before they got married). They typically were secretaries or did some sort of office work. The more career minded ones who went to college became nurses or teachers. Many of them were very smart. If they had been born year later, they would have been lawyers or engineers or gone into business.
If you look at a lot of the smart women now, they choose other paths. It essentially has been a brain drain.
The slack was not taken up by equally qualified men. I suspect that the perceived low salary had something to do with it. I suspect that fewer men have the desire to teach and nurture (and yes, I think a good teacher has nurturing qualities in the K-12 level). I also think the perception is that teaching is a woman's field.
The culture has change dramatically as well. Teachers (and administrators) seem to be in constant fear of being sued or berated by parents. I have a few teacher acquaintances and they say there is subtle pressure to pass everyone to avoid conflicts. Teachers also seem to be severely restricted in how they can teach which limits creativity. I think the focus on testing has gone overboard as well.
Unions for state employees were forbidden by law in most states until the 50s or so. (Even FDR thought have government workers unionize was a bad idea).
I am not sure having kids drop out is such a good thing either. While they may not be a drain on the teacher or the school, they become a drain elsewhere. The key there, to me, is to find a way to reach them and figure out what they can do and help them reach that. Not every kid is cut out for college, but can certainly get a certificate for training--there was a WSJ article recently about certificates. These differ from associate's degrees).
You are right that there were no vouchers then, but private schools were far less common (in the South desegregation saved the private schools and increased their number IMHO). Also, schools were segregated in many cases by law or how district lines were drawn. Not sure that was the greatest idea either. (How lines are drawn now can lead to the same situation although that would be controversial for sure).