Having left school 3 years ago, I can confirm there is still no grammar taught in english lessons. It was all taught to the AQA exam we did. I studied french and german too, and all my grammar knowledge comes from there and from languages at uni. There's obviously basic things in primary school but that's like where to put commas, that's it.

Historically we were taught Latin grammer disguised as English grammer - so 'tenses' when English only has two tenses (past and present) and 6 or more moods (shall, could, will, might etc, etc)

Most English 'grammer' discussed in the Anglophone world (split infinitives WTF? English doesn't have infinitives) is gibberish...

Well, now they teach made up nonsense about "fronted adverbials" ("Stupidly, they do it too early.") in primary school, they spend quite a bit of time on sentence structure and labelling the parts of speech. It's too much IMO and comes easy too early.

It's because English grammar is either obvious and doesn't need teaching or is inexplicable and isn't worth trying to make a pattern out of.

I ran, she ran, he ran, they ran, we ran, y'all ran.

I run, she runs, he runs, they run, we run, y'all run.

I'll run, she'll run, he'll run, they'll run, we'll run, y'all'll run.

I thought I'd run, he thought he'd run, they thought they'd run.

We'll have wished we ran, they'll have wished they ran, he'll have wished he ran.

What exactly do you want to teach here? The only weirdness to the rule is that he and she runs with an "s", everything else is the same.

I love you. You love me. They love him. He loves them.

I want you. You want me. They want him. He wants them.

In Russian the above is far different because the word order can be switched for emphasis.

The only thing in English worth teaching is "I have" vs "I am" and almost every language has that same problem as well (Je suis vs J'ai in French).

The hard part with English is that our spelling makes zero sense, unlike Russian or French. The grammar is really quite straightforward.

So verb forms are not the only part of grammar. In these examples you've covered sequences of tenses, mood, word order, and it could be argued aspect. Ask someone to point out the parts of speech and they'd be able to point out the pronoun and verb, that's usually it.

He runs to the store. I suggest that he run to the store.

Even verb morphology is a bit more complex than you've called out here.

A 10 word sentence has 10! possible word orders of which only 1 is correct - that is grammer

You seem to think that "verb morphology" is all there is to "grammar".

Also everything in English has dozens and dozens of exceptions that you just need to memorize.

A good way to teach that for native users might be to teach why that is the case and leave the specifics to happenstance.

This isn't unique to English.