I don't think most people equate being a career politician with being part of the establishment.
I see the establishment as those coming from established wealth and the Conservative party embodies this with over 50% of MPs having attended fee paying schools with 20 of these having been to Eton.
Life in school for these kids was a level playing field on grass.
I don't think the left tell their kids they are limited by their background. Many have come from limiting backgrounds and have got through so they feel nothing but ambition for their kids.