If the socialists had not fucked it up
The largest number of grammar (or "grammer", as you consistently write: I always mistrust people who can't spell the educational policies they are advocating) schools were closed under that noted socialist Education Secretary, Margaret Thatcher. It was a popular policy under the Tory government of the time, and neither Thatcher nor Major, both of whom had good working majorities, made the slightest suggest that it should change.
Let's suppose, just for the sake of argument, that grammar schools were a major engine of social mobility after the war. There's no particular reason to believe it's true (you can make a much stronger case for national service, and certainly for the war more broadly) but let's suppose. The grammar schools as we conceive them were a product of the 1944 Act, so the first cohort to go through them was born in the mid-1930s. As we're being accepting of your argument, let's suppose it has the effect of lifting a significant part of their 20% of the population into middle-class jobs.
Now, roll on a generation, into the early–mid 1970s when that cohort's children are going to secondary school. Who's going to get the grammar school places? Well, disproportionately they will be the children of grammar school parents. Assortive mating is a thing, but not every grammar school boy will have a grammar school girl (for a start off, there are fewer of them) so there will be a large number of households benefitting from the fruits of education and keen for their children to have some more of it. Are they going to get their children into grammar schools? Yes, disproportionately. They value education, they can deliver some of it and, crucially, they're willing to put up with the school being a bit further away and have working patterns which make that OK. Are some children from other backgrounds going to get in as well? Yes, they are. Some of them their parents will refuse to let them go (very much a thing) but yes, a few children whose parents left school at 14 will manage to compete and get in. But rolls another generation, and the effect will be stronger still. And if, by some mischance, the child of the middle-class family don't get into the local grammar, they will in many cases go private (which was, in real terms, cheaper in the 1970s): the glass floor to which someone alludes upthread.
That's why the abolition of grammar schools, or more precisely the improvement of secondary modern schools, in the mid-1970s was so popular. The middle-classes were frightened of secondary moderns, and the working classes realised they weren't going to be getting in and saw the consequences of it.
And that's also why, outside the backwoodsmen, no serious Tory who wants to get elected pushes for the policy. Grammar schools appeal to a certain ageing demographic, who hanker after a lost past. But any policy which says "jam for the 20%, bread and scrape for the 80% to pay for the jam" is electorally doomed, because most people are sensible enough to realise it's not in their interest. And the places where such policies are popular? Everyone votes Tory anyway, so there are no incremental votes.
"I wasn't going to vote Tory, but now I know they want to bring back secondary modern schools I will" said no-one, ever.
Grammar schools only worked, if they did work (and contrary to your simplistic claims, there's no clear body of evidence) because society was undergoing massive change at the time, and a wide range of middle-class jobs were opening up. Social mobility isn't zero-sum, but the economy grew and changed at a much sharper rate in the 1950s than at any time before or since.
The glass floor effect is very real. I can pay, both in folding money and in my time, to achieve educational success for my children, and I am motivated to do so. That value is passed on. There's a bloody good reason why grammar schools have FSM rates of around 1%.