As a text book, I have always relied on Fowler - whether he is suitable for people for whom English is a second language, I don't know.
By Fowler, I assume you mean A Dictionary of Modern English Usage by H W Fowler (1926). (Although there are later editions edited by folks other than Fowler.) As its title implies, it is a book on usage and not grammar. There is also, the lesser well known today, The King's English by H W and F G Fowler (1906). It has more grammar in it, but still has a lot of what I call style (usage). I personally like Fowler (I was introduced to him in high school), but his usage dictionary is getting a little long in the tooth. It's nearly a hundred years old and English, even British English, has changed greatly in that time.
Ill-educated people tend to speak the language (dialect) they learned, and they speak it grammatically. And, I agree with Goofy: children learn almost the entirety of their first language grammar by age 7, which is pretty much before primary school. What they may learn in school, if they are lucky enough to go to school, is a tiny subset of the grammar of the privileged dialect (i.e., the standard language) of their area. I don't watch much sports, but I don't know if I want them speaking formal English.
Again, I am asking for English grammars for teaching native speakers of English the standard.
OP
