It would depend absolutely on the school, the child and the area.
There is no private school within non-boarding reach of where I live that is better for my son (in terms of GCSE and A-level results, and top university admissions) than the local comprehensive. The local mixed state grammar is better still, but insanely selective (c. 1 in 1000 of the total Year 6 cohort each year get in). So for him, it's a definite 'no'.
For my daughter, there is one school - a nationally and internationally known girls' school - that is very very slightly better by those measures than the local comprehensive, but still behind the mixed grammar. However, the academic advantage is so very slight that again I would not consider it
If I were looking for SOCIAL advantage ('mixing with the right people'), rather than ACADEMIC advantage, there is a wide choice of private schools locally - but as I do not value social advantage, then no I would not consider them.
However, if my children had a particular SEN, or an exceptional non-academic talent (e.g. music) that could not be suitably catered for in a local state school, then under those exceptional cicrulstances I would consider private.
Equally, if we lived in a very different area, where my children's special needs (of being very academic, and in DS's case on the autistic specutrum) were not adequately provided for, then I might consider specialist private schools that met their needs. In the same way, my parents entered me for a (100%) scholarship to an academic girls' school when faced with the only other alternative, of sending me to the local ex-secondary modern (total O-level passes in the year I would have entered = 8. From 60 children).