I did my BA, my MPhil and am now doing my DPhil at a RG Uni thanks to Uni for having such fucking obvious classifications. As someone has said before me: the RG title and the power some seem to think it brings is utter bullshit usually only believed by self righteous mothers: "well my son got into Leeds so good fucking luck at Canterbury".
Far, far more important is the mark you leave with (and this is relative, you might want to further advance in academia, you might not) and, more essentially, the aptitude of the individual student.
For example, some people who attended non-RG & non-redbricks for very difficult subjects such as Law complain that they struggle to find employment after they graduate.
Lots of those people complain that it is because of the establishment they attended: that if they went to a RG they would have snapped up employment instantly upon taking up their mortar board. No. The likelihood is that they just found a better candidate than you. Perhaps the easiest way to accept this is to say "I knew I should have gone to LSE" and curse at how much more stupid their ex-classmates were than them.
I have made an error in my own education that many others at Oxbridge also commit: I have stuck to one institution for my entire educational career. When I graduate, I will have spent the best part of 10 years learning in the same institution, using the same methods, studying with the same resources. This shouldn't affect me greatly, because I hope to stick with academia. But, for employers, this is Very Bad.
Modern employers want to see flexibility, adaptability, the ability to cope with change; different institutions; different tutors; new places; different learning styles. Provide your child with a varied education, make sure they aren't boring let them have and build extra curricular interests and become independent, successful adults.
Shit that went on a bit. Sorry 