Fio tried "interview," twenty," and "winter" out in her own mouth and realized what the so-called pronunciation expert is hearing is NOT the omission of the T but the de-aspiration of it.
As Wikipedia explains, "English voiceless stop consonants are aspirated for most native speakers when they are word-initial or begin a stressed syllable . . . ," which means that sounds like T, P, and K are produced with a little extra puff of air at the beginning of words or a stressed syllable--but not when they don't have a starring role.
For a more complete discussion, Google Leigh Lisker. But take it from Linguistics Lady, Lisker ain't a-goin' ta tell ya that "ten" causes T to be silent.