Abolishing private schools would simply increase the size of education budget with no advantage.

The advantage is that it would remove a large inequality from the education system.  When those in power have to educate their children in the same places as the rest of us, then the standard of education will rise for all of us.  The money that rich people plough into unfairly benefiting their own offspring, is taken out of the wage packets of the poor.  That's one of the reasons they require their large pay rises to the detriment of lower paid workers.

What you need to do is to ensure that universities do not differentiate in favour of private schools

If there were no private schools then you wouldn't need to do this.  What you're suggesting is a sticking plaster to the real problem. But I agree with you that, to address the inequality in our society, we need to make inroads into the inequalities at all stages of education.  A child who has achieved three grade 'C' A levels from an inner city school has probably achieved as much as a child who achieved three grade 'B' A levels from a comprehensive school in a wealthy area.  And both have achieved more than someone who gets three 'A's from a top private school.  Universities should do more to assess the circumstances of an applicant's education and skew entrance requirements accordingly.

and there is (so far as I know) only one state school with a boarding house... - at least we were told when our kids were there that it was the only one!

No reason why this could not be changed though.