I just want to say that in my post I was talking about the biggest divide being between inner city and the wealthier 'burbs/exurbs. I believe a lot of that has to do wtih white flight. I was very fortunate in that (after my parents divorced), I was able to attend a very good high school. It was in what we now call an inner suburb. As white flight, at least where I am from, has continued past the suburbs and right out into what used to be farm country, those same inner suburbs are are struggling now, as well.
When I was younger (junior high), I lived in what was quite a distance out from the city. There were still farms and country roads. It was mostly blue collar folks. Many of the folks that had been there are now gone. It has become a wealthy *exurb*, complete with McMansions. My parents old land, 4 acres (sold after their divorce) has one sitting on it and is surrounded by them as far as the eye can see. The entire area has become that. They even had the name changed for the location. We lived on what was called a horseshoe. There were trailers on many, not all, of the plots. One of the first things that the wealthy did after arriving was enact a regulation that there could no longer be trailers. Gradually, they bought up the surrounding land. The school that I attended was pretty good, relatively new, surrounded by the elementary and high school. They still exist but now there are also state of the art, best in class, college campus type schools geared toward those new residents, closer to my old home. So, now those older schools serve the population of the small town while those new, excellent schools serve the population of the McMansions.
When the white and wealthy folks move further and further out, they take their money with them. The rising taxes tend to get those pesky previous residents out. When they do that, they really, really do not want to pay taxes that go toward wherever they left. Of course, I am generalizing. People are varied and not all are like that. In addition, it's not always just those exurbs who do not wish to pay any more than they already do toward schools, but they tend to vote against any state money going toward schools (aka voting Republican).
That said, yes, there are poor and/or disadvantaged small communities, as well, or even sections of the very same town. I currently live in a medium-sized town. Its schools are relatively good from what I understand, but they are aging quite a bit. There is much good about this town. I love it. It is considered relatively liberal-minded and open minded, likely due to having a large college population, though colleges being what they are, not all of those students vote here. The remainder of the population tends to be on the greying side. It voted blue (just barely) in 2016. It is largely completely surrounded by a sea of red. Recently, there was a referendum so that the older schools could fix their very aged HVAC units. It would have increased property taxes. It was voted down.
Many people may be struggling, no matter where they live, and do not wish to have their property taxes raised while they are already on the rise to support schools, libraries, what have you that have lost their state funding. It tends to be more pronounced in inner cities where there are already more lower income folks who can least afford those rises in property taxes and of course, their landlords who don't want to see a rise. But still, many folks in those same areas will vote for a rise in them when it comes to their schools.
Property taxes are a horrible way to fund schools, especially as a certain party will defund education so that becomes the way that schools have to survive. Poor stays poor, wealthy stays wealthy. I will die on that hill.