This is speculation, but I wonder if the period of emphasizing explicit grammatical instruction wasn't an accidental interregnum. That is to say, back in the days when Latin and/or Greek were part of the ordinary curriculum students learned grammar much as I did, as a "natural" excelerant to interpreting a foreign tongue. Once those languages were dropped educators noticed students couldn't do grammar analysis anymore, and so tried teaching it directly, without fully considering when and why it might be useful. I don't know how well the dates line up, but it would be interesting to look into.
This is one of the reasons why Latin is tought. You learn transferring a gramatically hard language into your own, having to learn the ins and out of your own language's grammar. No grammatically complex situation in your own language can fluster you afterwards.
Best class I ever had!
It's so widespread that today if you want to play word guessing with gestures, and you have several words, you just imitate that underline style, and everybody understands it. (Just remembered, we also did a lot of word analysis, marking up prefix, root, suffixes and ending, and everyone knows this markup too.)
I won't really agree that mastering grammar of native language limits on how well you can learn other languages. Maybe it matters in the way how it taught in college, when you are older and approach to learning language is "more structured". But when I learned Georgian at age of 6 and Hebrew at 12 (through very deep immersion. Teachers spoke only Hebrew), English at 14 (I had 5 months of private lessons following by dial-up connection to mostly english internet), it didn't matter. At least not for me.
There was also this interesting phenomena, that immigrant when they went to local school, their scores in hebrew grammar classes were usually higher than those of native speakers.