Not necessarily true. I live in Forsyth county GA. It's one county away from downtown Atlanta (Fulton county) but is still part of the metro Atlanta area. When we built our house 17 years ago it was the fastest growing county in the nation. My property tax is around 60% of what I would pay in Fulton county which is only 3 miles away. Our schools are FAR better than Fulton county and most other schools in the Atlanta area and the entire state. My house is on a bigger lot yet costs 20% less. We have far less crime than Fulton county and better infrastructure. The tradeoff is more traffic if you have to commute to downtown.
There are many reasons why people want to live in a particular area and it doesn't always mean less crime or better schools. It has far more to do with image and access to infrastructure, public transportation, restaurants, night life, events, etc. It also has to do with availability of new building sites.
Living in expensive areas is a lifestyle choice, not a requirement to be safe or get a good education.