In my own experience, those kinds of triplet quarter notes or half notes tend to be wonky in classical ensembles -- even professional ones. Same story with quintuplets.
While the video is massively over-generalized, I actually agree with the larger point that classical training tends to under-emphasize strict rhythmic precision. That's especially true for string players, and doubly so for singers. That has a lot to do with the standard repertoire we're trained with, where complex or tricky rhythms are uncommon and not a major feature of the music. We place much more value on tone quality and phrasing, and having a somewhat elastic sense of time is actually essential for much of the classic repertoire to function.
It's a big jump for most classically trained musicians to play other genres, including most contemporary classical music (which often does feature complex rhythms and precise timing). And as for the original question, in most contemporary classical ensembles (mine included) it is essential to play exactly with the conductor's stick.
I've gotten so used to that, in fact, that when I do play with orchestras it always takes me a few minutes to stop playing "ahead" of the beat.