How much does knowledge of linguistics help with learning foreign languages?

[FlatAssembler]

My (very limited) understanding of linguistics is that languages are grouped moreso on syntax and grammar than phonetic similarities. Phonetic similarities are more a symptom than a cause. I'm happy to be corrected if I'm wrong on that.

The way languages are proven to be related is by finding simple regular sound changes and finding many examples of them in words that are unlikely to be loanwords (numerals, body parts, basic agriculture…). Sound changes I've listed would be examples. Of course, linguists would try to find as many examples as possible, much more than I did, to make it less likely that those examples are due to chance alone. Grammar can be used to prove language relationships only if you can show that regular sound changes operate in grammatical words (like the conjugation of the verb "be") or, rarely, in affixes (like conjugational endings).
And, yes, I think the schools place too much attention on grammar when teaching linguistics (I also think they are teaching grammar the wrong way, but that's another story), and way too little attention on the laws which govern how languages are related.