Property taxes do pay for education - corporate property taxes especially are what fund most of education since our fair government continues to pass initiatives without funding them.
I live in an area that has high property taxes. There are additional taxes charged by the school district. I moved here deliberately and am happy to pay them. My children will go to a public school that is known nationally and has name recognition with major universities. A full 85% of the 3,000+ students are university bound. The average across the state is closer to 25% of students in a high school.
It works like this: High property taxes, especially school district taxes, generally correlate to a desirable school district. Those who live there vote the higher tax rates in to improve the schools. Local public schools are one of the most important factors in home values. People who value education and family want to live there, so they pay accordingly. Despite the "housing slump" home in my area are still gaining in value and sell in less than a week thanks to the schools. This is not true only a few miles in any direction.
If you don't have kids, and don't want to pay for a powerful school district, don't live in one. Of course, you can expect your neighbors to be childless (which is probably desirable to those who don't understand children) or to be the kind of parents who have no clue as to how, don't have the means or don't have the desire to provide the best education for their children. BTW - in most areas, private school teachers are less qualified than public school teachers, you can thank the government for that one as well.
The quality of public schools has some to do with the quality of teachers, but much more to do with the quality of parents. If the parents of students within the school demand excellent teaching - it happens. If they can't get their shit together to find a "good" school with "good" teachers, teachers will do the best they can or do enough to get by since it must be good enough to satisfy the parents and other adults who have no clue how public schools really work.
Schools are a business that can not reject a single piece of inventory - no matter how flawed it is. Public schools are one of the few institutions that have this requirement and they do the best they can. The quality of the school is a product of parental support, "raw materials" and funding to meet all of the demands of the government and parents. If parents make demands they are usually pretty good about paying for them, i.e. higher property taxes.
The rest of your property taxes I know relatively little about, but I know a bit about how public schools work.
Oh, and my children don't cry in restaurants either. Again - look to the parents and the quality of parenting in the area you're in.