Oxford and Cambridge have always counted (at least part of that was old boys network, in the same way private schooling was).
In the days when a distinction was made between universities and polytechnics, there was definitely a snob perspective. Polys had less stringent admission requirements.
Since all the polys became universities, the distinction isn't there, but what we have to bear in mind is that far more young people have degrees now. It's not a fair way of doing it, but it is a method of selection when faced with a bunch of candidates with the same qualifications on paper.
It's like GCSEs. Pretty much after I left school and had got my first job, subsequent employers weren't so interested in my exam results. They'd ask for the number of subjects I had, but I was rarely asked to prove it. After I went back and got my degree, that really was the end of any interest in my GCSEs.
Now, though, with so many kids coming through with degrees, employers are looking further back to help with selection. Whether it is or is not really an indicator of ability in the job market, it isn't the distinction it once was. When I did my degree in the 90s, only 19% of a cohort went to uni. And that was a lot in comparison with the previous decade. Now, over 50% of a cohort will go to uni, which means the little distinctions matter more, especially in competitive fields.
Once you have been working for a while, it will matter less, because prospective employers have something else to go on.
When we leave full time education, all prospective employers have to go on is our exam results. It's a buyers market at the moment so results matter now more than ever, and where you got them from, in a sea of peers with the same qualifications, where you got them is, sadly, going to be a factor.