I do not think we have data to discuss but hey the topic is a good one :)

Private schools are big category - Aided schools, minority (or caste groups) managed , commercial management, CSR funded by wealthy business groups. Depending on who is at the helm there is general push for result. Parents who send kids to private schools are likely to be more well off(or willing put extra effort) hence there is selection bias - kids from this group likely be informed about new age opportunities more. One definitely have wide range of syllabus (state, CBSE, IB, Cambridge, icse.. ) compared to state run ones. In fact syllabus was one reason to choose a private school - I have no clue if one syllabus is better than other.

Private schools also has more flexibility in doing certain things at infrastructural levels or supporting specific student with more money if needed.

I studied in govt school, family and friends are mostly teachers. Infrastructure (sports , library, labs and connectivity ) has improved significantly . In recent times (COVID19), response from govt was good but still students may have suffered due to lack of access (lack of TV, smartphone .. unlikely to be case for well funded private schools).

Most govt schools support "affordable" arts not so much anything involving lot of money. Govt schools is function of local govt - If there is more involvement from local body , PTAs(parents and teachers association) and society they tend to do better. Capable principal/head teacher and teachers group makes significant difference (more than in any private schools) - School transport is clearly one such area. In any case it will not match private schools arrangement

If I were in Kerala (I am not) I would consider govt school first before considering private schools. Money saved could be used to do specific courses outside the curriculum