Most schools do not follow the rule, please Google - Tamil Nadu Tamil Learning Act, 2006 (Tamil Nadu Act 13 of 2006).
If your child comes to board exam stage and does not have Tamil, he may not be eligible for government seat under NEET. There is generally a lot of confusion and ambiguity about this but you need to find a school who teach Tamil as a part of their subjects till 9th in 2023 - 24. Whether Board exam in Tamil will be mandatory is unknown now.
Hope this helps.
You are welcome Anupa, please decide whether your child should study in India / abroad for UG and then decide which board to go for.