Of course, I would not translate "F#" as "F sharp", that was only meant to illustrate what might be written in the english transcription subtitle. But the english transcriptions do also vary between the two forms, so some flexibility is certainly needed.
About the "literal" form, where just each "symbol" is pronounced as is: as you say, it is not really correct, and I think it does also not really help to get a better understanding, especially as the compact notation is usually not given. So in the video, you will hear Justin saying something like "C minor seven sharp five" (which is, as you already mentioned, pretty much just gibberish for those who have to rely on subtitles) and the subtitles would read "C moll 7 kreuz 5", which is not really the correct translation and thus, does not help much to read the notation better. And like you said, if you are at a stage where more complex stuff is mentioned, you can just read it like that yourself even if it is written as Cm7#5.
So I think we can agree on using properly spoken names in beginner-style lessons (maybe together with the notation in parenthesis) and maybe where it is compact enough and using the proper, compact notation everywhere else.
In general, it is kind of impossible to just learn everything from videos, especially notation (I don't know if Justin even has a video lesson on that topic or on pronounciation "rules"). So there is only so much we can do to translate those things.