I think calling it a wealth gap really misses the point - it's a stability gap. Most upper-middle individuals pass on socio-economic stability to the next generation. Most middle and lower class pass on instability. Wealth is a small part of it.
I was agreeing with your comment until the last sentence. Wealth is stable access to money, which is a fungible resource. Used appropriately, it can provide access to stability that would otherwise be inaccessible. Without such a resource, other factors of instability more likely to be chronic or even inescapable.
Education is actually much more important than wealth. So is city and community infrastructure, a positive family environment, etiquette / upbringing, values passed down, positive / helpful relations, mentorship, trauma free environments/mental health, and good social practice. Wealth is a very small part of stability.
