The short answer to your titular question is no, wrong.
As for your first source video, I hear a space between the definitely pronounced
/s/ on the end of carrots and the beginning of the next word, so as far as I can tell the speaker doesn't say
/ˈkærət suː/. Listen again and see if you can't hear him say something more like
/ˈkærəts tsuː/. This is closer to how I seem to hear it. In any case, I think he does say the
/ˈkærəts tsuː/ is what he says, it's more likely a momentary failure to enunciate properly, perhaps influenced by the cluster of sounds involved. Maybe the t on the front of too comes out as
/ts/ accidentally, because of the
/ts/ on the end of carrots. (There is also, perhaps, the possibility that an individual tendency, a peer-group tendency, or an artifact of the recording is at play, or, to put it another way, it probably can't be completely ruled out on the basis of this lone video.)
As for your second source video, the
/t/ is pronounced there too. You just have to listen better.
There is certainly no pronunciation rule that allows the
/t/ to be dropped in circumstances like this. Generally both the
/s/ and the
/t/ must be pronounced where st is present.
Note, however, that there are a number of words that do not have the t pronounced after the s where they end in stle, e.g. castle, whistle, and hustle. This could perhaps be considered a rule of thumb. However, even this isn't an absolute rule for all similar words. Pestle, a word with the same last four letters, is an example of a word that is pronounced with the
/t/ by some, and without by others.
And this dropping of the
/t/ between s and l does not work between two words (even if those words are pushed together to make a contraction). So pronounce the t at the end of cast in The cast'll be coming out to take a bow, and at the end of past in Looking at the past lets us more easily predict the future.
This answer doesn't take into account all varieties of English, but it does apply quite generally, and it accords with the English spoken by the speakers in both videos.