"For example, making the entrance exam tutor proof, actively targeting pupils on fsm etc"
I think it has been pretty well established over decades that there is no way to make these tests "tutor-proof," and it is not just about tutoring, in any case.
Everytime a well-heeled middle class family reads to their child, listens to their child read, takes them to a museum or art gallery or stately home or concert, brings them to an extra curricular activity, bathes their child in language and stimulation, puts their child in an enriching holiday club, takes them traveling and shows them the world, they are boosting their child's academic smarts and therefore (ultimately) their ability to pass testsany kind of testsat the end of primary school.
Inequality gets baked in VERY early.
It is not just about tutoring.
Even if the state threw up its hands and starting actually funding free private tutors for poor kids (and can you imagine how much that would cost?), you would still not get even close to getting rid of the entrenched disadvantage that has built up in poor kids by the end of primary school.
Grammar schools are a scandal and the data from Kent and other places shows clearly that that they worsen outcomes for poor kids.
We need strict (but caring) academically excellent, high-expectations secondary schools (and primaries, too) for ALL kids.