That is true. If my daughter hadn't gotten into a good charter school in the city, I wonder if we hadn't looked a lot harder at moving. However after we did eventually hop over the county line my daughter still 'commutes' back into the city because she doesn't want to change schools.
Suburbs can restrict their schools to those that can afford to live there, so those public schools can have very few kids from lower socioeconomic classes, and a greater proportion of kids from people who are in higher earning professions.
I grew up middle class in a upper middle class school district, with attentive and loving parents. My mom stayed at home and raised us. I was still exposed to skipping class, drugs, and alcohol in middle and high school. I almost didn’t graduate myself, due to poor choices. This is in a school with a 92% graduation rate. I understand from intimate personal experience that the more opportunities your kids have to interact with peers making poor life choices, there is a higher chance your kid gets caught up in it despite what you do as a parent. This isn’t sheltering (I certainly wasn’t sheltered), it’s just repeat exposure to exciting but bad choices leads some kids to temptation.
I could have moved to a more expensive neighborhood in the city but the same fundamental problem exists. I don’t care about how much money people have, I care whether in aggregate, the parents in my community try to raise their kids with intention. There’s a shocking lack of that in 40% of the families in the schools in the city we moved from.
I'm familiar with the argument that the cohort of students matters more than the quality of instruction. in fact, that is what I'm getting at here. the city has an explicit policy of mixing students from different backgrounds across the school system, the goal being to break the cycle of poor students going to poor schools and staying poor. it's a good goal, but it seems kinda pointless when all the students they are trying to mix them with are fleeing to the county.
Sometimes it's the school, not the school system.
at least in my area, I wouldn't say the city has "abandoned" families. rather, it has focused mainly on improving public education for children from poor families. this is a noble goal, but it imposes tradeoffs that the middle class families are mostly unwilling to accept. since they have the ability to leave, they do.