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The problem with that approach is that adult second-language learners generally don't put in the same amount of time or access the same breadth of material as a baby learning it's mother tongue. Every word has a frequency with which it appears. A child will come across almost all words a sufficient number of times to eventually memorise them, because they are totally immersed in the language every hour of every day. As an adult learning a second language, unless I'm living in the target country in a fully immersive way (which for me isn't the case), then by consuming 30-60 minutes of media a day (which is my upper limit of what I can realistically achieve) I'm going to get strong exposure to high-frequency words and grammar patterns, and weak exposure to low-frequency ones. Many of those at the bottom end will be so weak that the occasional exposure I get simply isn't enough for me to attain fluency with them. Anki solves this problem: all words you learn via flashcards have a (roughly) equal chance of being remembered, independent of their frequency.

Anki also allows you to take long breaks from learning. If I go a year without learning any new material, provided I keep up my reviews (which significantly diminish in duration the longer you go without new cards), I'll pretty much be able to pick things up again where I left off. That doesn't work so well with other methods because you will forget a lot.

Flashcards are a lot more efficient in terms of number of minutes spent per word. For example, Skritter tells me that I spend an average of 1.78 minute learning how to write a word with Chinese characters. Aside from the fact that I wouldn't practice writing at all if I just consumed books/films, I'd also spend a lot more time that way as I'd constantly be stopping to look characters up in a dictionary and/or googling the grammar points every time I forget.

I take your point that you find Anki boring but that's highly subjective. I actually find it very satisfying and rewarding, almost as if I'm downloading information into my brain Matrix-style (just slower). There's a sense that whatever knowledge I put in Anki is mine to keep forever. In ~10 years of language learning, Anki is the one thing that I've most consistently kept up with. Your claim that it's a "weird cultural instinct [to make everything] as dry and uninteresting as possible" is false consensus bias - you're projecting your own feelings onto others who don't necessarily share those feelings, and therefore assuming everyone else must find it boring too.

Anki is also more suitable for beginners than books/shows. Realistically, you can't read a book or watch a movie when you are just starting. Everything will be so incomprehensible that the effort of having to stop to look things up will be overwhelming and tedious. Anki on the other hand can be started from your very first word or sentence.

For me personally, I neither like to dismiss nor focus too much on any one method. I've always learnt best when I put effort into multiple different methods: Anki, books, audio, apps, TV, real life practice, etc. This also helps to keep things fresh and interesting.