In some countries students do not have access to native speakers. In such countries the grammar-translation method still dominates. Emphasis is placed on reading and writing, rather on listening and speaking. This method is still popular in Europe, especially in Eastern Europe, where English is just another academic discipline.
If a teacher regularly tells his/her students anecdotes and stories for enjoyment, students will pick up a lot passively. Try it. Tell your students an interesting story just for fun. In the next lesson ask your students how much they remember. Not only will they remember the plot, but they will also retell it more or less with the same vocabulary and grammatical structures.
Maybe the issue of too much TTT is probably about the teacher chatting too much because he/she may not have prepared for the lesson. But hang on, isn't small talk good practice? The most perceptive student will pick up a lot of language.
Maybe some teachers should stop looking at their students as empty barrels waiting to be filled with information about grammar etc. Students come to the lesson usually with knowledge of two or more foreign languages, life experience etc.
And anyway, did your parents give you formal English tuition as a toddler so that you could speak English? No, your parents talked to you and you just babbled back and you probably made loads of grammatical errors as well.
I have a 13 year student who picked up English from MTV, CNN, Cartoon Network and rap music. Lots of students pick up language in this manner. Likewise, students can also pick up language from a native teacher.