"Grammar schools were intended to raise the social status of poor but bright children who would have found it difficult to better themselves otherwise."
That may have been the intent at one point, and it may have been the effect in some cases, but it probably wasn't the case post the '44 Act. The Golden Age that's usually invoked to justify selection as an engine of social mobility starts with the post-war generation who went to university in the 1950s on the back of the '44 Act, who would have been lucky to have had a complete secondary education, never mind a university education, a generation earlier, and ends in DES circular 10/65 which brings in comprehensives in the light of the Leicester experiment. The trailing edge of that generation is David Frost and TW3, the leading edge of that generation is my parents (cards on the table: my parents passed the 11+ in 1946 in the immediate aftermath of the war and the '44 Act, so I'm an indirect beneficiary.
The problem is, in all the post-war reconstruction, the idea that grammar schools might offer social mobility was handily glossed over. If you look at a map of a city with substantial post-war rebuilding, either as a result of bombing or slum clearance or both, you'll find that the schools built after the war on new social housing developments were mostly secondary moderns; the idea that the working classes (who were the main inhabitants of those new developments) might pass the 11+ was almost alien to the planning. I looked at the rates as part of my 16+ (GCSE pilot) History project in 1981 of which I unfortunately don't have a copy, but my memory is that the proportion of grammar school places in newly developed parts of Birmingham was about a third that in leafier areas, and it is routine to hear of people today whose parents passed the 11+ but went to the local secondary modern because the grammar school was logistically impossible.
If you know Birmingham, look at the housing centred on the former Longbridge factory, and consider where the nearest grammar school was to there --- in 1955 the answer is Kings Norton Boys (Five Ways was still at Five Ways). That's why the policy arose until the abolition of the 11+ in 1974 that if you passed the 11+ you could go either your local grammar (of which there may not have been one) or the nearest of the prototype comps. The early comps (Great Barr, Shenley Court) had stellar results --- multiple Oxbridge each year --- up until 1981 (ie, the departure of the last cohort to take the 11+) and were seen to "fail" later because it was only after 1981 that they had comprehensive intakes emerging at 18. Some grammars were of course built after the war, and some were around new-ish housing (Bartley Green Girls, now Hillcrest, for example) but they are few and far between compared to the secondary moderns.
My impression is the same goes for most other big cities: the pre-war grammars persisted, usually in affluent areas, while the new housing built to replace either war damage or demolished slums was built pretty much exclusively with secondary moderns. Few authorities were willing to put up three schools to support the tri-partite system, so they put up one, which was inevitably a secondary modern (and funded as such, and lacking as sixth form as such). There were some honourable exceptions --- the reason early Birmingham comps have lower, middle and upper schools as separate buildings, with a distinct workshop block and lots of halls and gyms was so that if comprehensive schooling failed, they could be converted to tri-lateral schools on one site. And Arthur Terry was built by Warwickshire as a bilateral, combined but separate-ish secondary modern and grammar on one site, and wasn't alone in that. But in general, the rate of building secondary modern places far outweighed grammar places, and disproportionately so in areas of deprivation, so the grammar schools became progressively less egalitarian unless people were motivated and able to send their children a long way to school (this pre mass car ownership, of course), which is pretty much a self-selecting group.
I haven't looked in recent years, but I recall seeing studies of the social make-up of grammars in the 1950s and 1960s, and they were distinctly right-slewed compared to the secondary moderns. And the pass mark was lower in more affluent areas, because of different proportions in the provision. It wasn't a golden age.