Well, the whole thing is rather counterproductive—a lot of time wasted on matters of very little importance. However, I'd say that the AmE speaker is not dropping her 't's in that segment of the video; she is merely stopping them. If we had a chance to listen to those two speakers in natural conversation instead of self-consciously pronouncing isolated words and phrases, we would find more similarities than differences, I suspect.
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Ruslan L.... at the end of the word or the syllable the "t" is dropped - cut ...In American English some Ts are unreleased, not dropped.