Is Grammar really that important now?

+Dimethylamine

I’m not sure if discussing this further is worth the effort.
Let me point out (again!) that the expression “study grammar” in the context of the OP’s question is whether he should learn the rules by heart not whether he should completely disregard concepts.
We’re basically on the same boat in practice and, as I’ve said, I myself has gone through all Russian declensions, grammatical quirks, syntax, perfective vs. imperfective, …

However, just because you seem interested in the discussion and just to clarify the ideas, I’d say that I’m not convinced by your example
You can say the same about any language, it’s only a matter of whether the particular structure present in your target language is or not familiar to you.
To follow your example. Confronted with that situation, a German speaker would think “oh, so тебя is dich and тебе dir” I see. No need to know “grammar” in the sense that you’re implying. An English speaker might think, hey! Why is it sometimes теба, sometimes тебе? You may want to include some grammar lingo to explain it (dative, accusative, DO, IO) or not, just use examples
Well, if I give a present to you you must say тебе, otherwise it’s as if you were the present, …

Again, you can say the same about, for example, English
I give you a present
I’ll give you a present

So, what the heck is that “l” sound used for??? Again, if your native has a future tense, you may think you don’t need an explanation, whereas you may want one if your language doesn’t have it. And again you may provide it in grammar jargon or with examples. It’s exactly the same, English needs as much “grammar” as Russian. Consider all the tenses English has and Russian doesn’t and don’t get me started about articles, which is much harder than any dative/accusative distinction.

However, most people think that learning rules to use the articles doesn’t make sense: you learn through example. Declension is just the same! You may want to look more or less at the rules but you’ll need the examples and exposure or you’ll never get it right! How much use you make of the explicit rules is up to you: you can decide you like learning by heart or at least reviewing tables from time to time; you may want to avoid memorising but have some idea that it’s a thing worth looking out for in advance, you may just go by the exposure. The procedure would be:
Be attentive (as Steve says), look for patterns if you detect something that’s unlclear теба/тебе, look it up, ask about it, get a feeling for it through examples.

That’s learning grammar in my book, it works for learning to use articles, subjunctive and declension and it’s always a possible alternative to explicit rule learning or a possible complement. The opposite’s not true: mere rule learning doesn’t make up for the lack of this kind of process

Again, again, again. I’m not saying that explicit rules don’t help: they may help a lot, they help me and I do use them but
a) they’re not indispensable, and they’re not in Russian as they’re not in English, it’s the same thing
b) it depends on your learning style and current just how useful they are for you at this moment

That’s what my answer to the OP tried to express