You are rather profoundly missing the point.
Social Mobility is defined in terms of the probability of somoene born into one economic quartile ending up in another one. Note the definition: It's a zero sum game. Upward social mobility requires downward social mobility, and the middle class dominance of second and third tier universities gaurantees that their children will rarely move down an economic rung.
The reduction in Social mobility over the last three decades isn't predominantly comming from the range of people attending the elite universities which provide the majority of places in the top economic quartiles, it's the range of people attending middling universities that make up the bulk of the expansion of the higher education sector.
"the he deserves to go to a good school"
And if he starts in a bad school that inhibits him getting good grades?
And why restrict the "good schools" to a limited percentage of the population based on a test?
It's not an impediment to others no, but that is not and has not been the point. I'm rather surprised that people seem to want to have a different discussion than the one we are having.
I am not arguing that Grammar Schools are an impediment to others. I am arguing that the decline in their number is not responsible for the decline in social mobility, that they will not increase social mobility in an economy when the middle economic quartiles are dominated not by excellence that Grammars aim for, but by mediocre jobs secured by mediocre degrees which neverthelss the poorest have impared access too due to the failings of the state education system. Therefore there is no political argument for "bringing back grammar schools".
The whole concept of Grammar schools is from a very different era:
Only the top x % of most inteligent people will go to university. Since then the proportion of the population attending university has more than doubled, depending on when you take your reference point. IIRC it was something like 15% in the 60's.
In that context, it provided social mobility as it opened up to the top jobs attained through elite universities to everyone, while the lower economic quartiles were relatively intermobile.
The idea of selecting a narrow number of state pupils for elite education just doesn't make sense in an era where we are expecting ~50% (and we are pretty close to that) to go into higher education.
Trying to make grammar school out as a reward for working hard is a poor fig leaf at best: the number of places is strictily limited, it can ONLY be a reward for a relatively small number of pupils. You may have grammar places for 10% of the sate educated pupils, and find 15% of the population making up a defined cluster of hard workers... so the top 10% of that 15% get "rewarded"... fine then, as a method for rationing scarce resource (good education), but it's still not going to impact on social mobility.
Ok, so we can expand on that, increase the number of Grammar schools. Taken to it's limit, ultimately, what you are saying is you want a fully selective state system, much like the indpendent system, catering for the dim through to the mediocre up to the brightest. Fine, I can see the merits of that, but you can get to the same result by streaming and setting within schools, with the benefit you also get the localism that many people want in public services.
Do we really need Grammars to achieve streaming and or setting? I don't see why.
Can you argue that Grammar Schools are needed to improve social mobility: Only if you intend ~50% of pupils to be attending Grammar schools, which is a long way from the orrigional concept of Grammar schools. If you remain with only a small limited number of grammar places, keeping it elite, you are only changing the educational experience of a very narrow number of state pupils, and it will have no impact on the actual measure of social mobility.
Fundementaly, the flaw in this argument that Grammar Schools Provide Social mobility is that you are looking at an educational model that is based around elitism, while we are working in an economy where the majority of people (the middle classes) are rewarded financaily for mediocrity which they can easily gaurantee, while the poorest tend to have to be excelent to move up economic quartiles. The statistics show that: the probability of moving into a lower economic quartile from the one you are born into, if you are middle class, is very low, and the price of that is that correspondingly there is a low probability of someone born into a lower quartile moving up.