The Sutton Trust has a lot of info about this. The disparities are already in the system. Adding even more rich parents to state education will only increase the existing divide. Look at this document, for example -
https://www.suttontrust.com/wp-content/uploads/2023/03/Tutoring-The-New-Landscape.pdf"[Students] at grammar schools were most likely to receive tutoring (23%). Many independent school pupils also received private tutoring, over and above the resources provided by the private school sector (19%). While the overall figure for comprehensive schools is 18%, this hides enormous disparities within the sector. At the least deprived comprehensive schools (by rates of Free School Meal eligibility), rates of private tutoring were 31%, compared to 12% at the most deprived schools.
Looking at household characteristics, in particular measures of socio-economic background, there are significant disparities, as you would expect, the most obvious of which is income. Those in the top quartile of household incomes had, by a distance, the highest rates of tutoring, at 32%, with a significant gap even to the second highest quartile (18%). This is in the main driven by extremely high rates among the top rung of incomes (net household income of £75,000 or more), of 35%. This contrasts with less than 13% of those in the bottom quartile of income paying for private tutoring. Though this figure is likely to be inflated by some well-off households who report low or no income.40 Looking at social class, as defined by the NS-SEC three category classification, 24% of those in professional and managerial households received tutoring, compared to 17% of those households with intermediate occupations, and 11% of those with routine and manual occupations (including those who never worked). There are also associations with the education of parents, with 26% of those with a graduate parent receiving tutoring, twice as many as those without a graduate parent (13%). Pupils in houses with a co-habiting partner were also more likely to receive tutoring (20%), compared to those living with a single parent (12%)."