We should try carrying out a RCT where we assign kids from deprived backgrounds in certain areas to take part in paid-for extra-curriculars, then compared their academic outcomes against kids in similar areas where no intervention took place.
My hunch is that it might improve their academic performances just a tiny bit (by encouraging confidence and some soft skills like resilience or delaying gratification etc.), but would not have any particular transformative effect on their scores in maths, English, history, geography, science or anything else.
In general, domain expertise tends not to be very transferrable; the evidence is that learning music (for example) has no real impact on attainment in other areas, including maths (contrary to what a lot of people believe).
That doesn't mean it would be a complete waste of time, as it might well be good for physical and mental health, reducing screentime, and just be a wonderful experience for kids.
But if the aim is "improve scores in maths, English, history, geography, science," then you need to do other things.