But TD, young children at home learn a language by osmosis/exposure: by hearing the words,
thousands of times as a baby and later as a toddler.

Several differences:
1) this is a second language; there have been studies done in families who are as fully bilingual as possible from birth. In those situations, the 'environmental language' on TV, in the store, etc. still induces a preference and a variable strength.
2) the role of 'nurturing' language, mom's, can never be recreated in a classroom with multiple students and distractions.
3) language acquisition of 2 year olds, 5 year olds, 8 year olds varies greatly, even by individual. A teacher in a classroom with even a hand full of students cannot duplicate a parents innate one-on-one responses and attention.

A teacher certified in elementary education in second language learning does have strategies to adapt to his students. But multiple language teaching in Europe (Piaget's home), with decades of experience, still struggle to generate true bilingual young students. A NES FT in a private school is simply marginally effective.