Teaching grammar is probably not the same thing as teaching the standard variant of the language. People can be told what an infinitive is without being commanded not to split it. On the other hand, they can learn to speak like the British queen simply by observing the queen, without being taught how to tell apart subject and object.

The purpose of the former is to promote understanding of how language works, and maybe to help people to structure their sentences and speak more clearly (if needed) or make foreign language learning easier.

As for the latter, the purpose might be dialect elimination, but I am not sure. I don't know how much exactly the local dialects are eliminated in France or Russia, but in the Czech Republic, where I come from, the local dialects are not eliminated. There is complete mutual intelligibility between all dialects (and almost complete mutual intelligibility with Slovak language and its dialects), but the difference between how most people really speak and the standard language taught in schools is substantial. I have read that such (or even greater) diglossia exists also within Finnish, all variants of Arabic, Greek or Welsh.

Dialect elimination was an important reason probably in France where language unfication was an openly admitted political goal and "patois" have had very low status, but I suspect not everywhere. (France still has problems with recognising minority languages and has probably the most restrictive language policy in Europe.) Since I have not been schooled in France, I have never been told in school that dialects are bad or that I should speak always the literary standard. The standard was intended mostly for writing and official communication (e.g. TV news) and in private communication people are expected to use whatever dialect they speak natively.

I think that importance of teaching a standard language is probably overrated, but still it does probably make language changes slower, which can be good since future generations will need translations of today's texts later (maybe not an important issue since we will have good automatic translators soon). People also tend to treat the standard form of whatever language they speak as a terminal value, which is harder to argue about.