But you first learn a language intuitively at home, by using it a lot and developing intuition for what sounds right and what doesn't.

I'm a native Polish speaker (which has a "typesystem" much more complicated than the German one nevermind the English one). I've learnt German for 4 years in secondary school. I remember the system to be pretty intuitive and simple (mostly you change the articles not the words, there's just 4 cases, no verb aspects, just 1 kind of plural, just 3 genders, etc.).

So it was definitely easy to remember the rules and I did at one point (not anymore ;) ). But still to speak German you can't pause every 3 seconds to do a table lookup for the correct conjugation and rules, that would be far too slow. You have to use the language and have that "ouch" feeling when you use the wrong case so eventually you know intuitively which word to use.

A game to teach a language should just be a game using that language and giving immediate feedback.