@PingvsPong
You out forward interesting points.
I certainly wasn't trying to denigrate teaching though I would argue the government has. The fact that you think possibly quite rightly entry into teaching doesn't confer social mobility shows how we need to fundamentally address the social and financial status of teachers.
So Mom and Dad are teachers and have a bright child that goes to Oxford and become a a teacher like their parents in a technical sense isn't a social climb but in terms of society I think it is a really good thing for various reasons.
Maybe we need more examples of Oxbridge grade that make real contributions to sectors that are in the public interest and not necessarily immensely renumerate or high profile.