Let’s state it like this: in English there is a distinction between the “p” and the “b” in two ways (although dependent on the accent): generally the “pa” is aspirated and always unvoiced, while the “ba” is never aspirated and in almost always voiced.
In Korean there is no distinction between voiced plosives and unvoiced plosives. That is to say, whether you use your voice or not usually does not sound different (or “wrong”) to Korean natives. On the other hand, there is a difference between aspirated and unaspirated (called lax), as well as a “tense” version of these plosive sounds.
Whereas English speakers would have difficulty recognising the difference between an aspirated, lax and tense form of an (unvoiced) “pa”, Korean speakers would have difficulty recognising the difference between a voiced and an unvoiced “ba” versus “pa”. But since the latter is usually aspirated in English, it sounds to Korean speakers as if the “b” is more correct.
Meanwhile, in many European languages, both the “b” and the “p” are not aspirated (much). For example in German (which I presume is one of your languages), Dutch (which is mine) or French. Here the difference should be hard to distinguish for Korean speakers, but easy for German speakers, for example.
It is therefore not weird that the person in your video uses a voiced “b” at the start of Bibimbap, but it would equally be not weird if they said it with an unvoiced “p” at the start (this difference doesn’t exist in Korean).
However, as far as I could gather from watching (carefully pronounced) tutorials on how to pronounce Korean (or at least the amount I remembered from it, it has been a year or two), generally these plosive consonants are unvoiced, unless they happen to be in between other sounds that are all voiced (i.e. sonorants). This doesn’t happen in English, for example the “p” in “compartment” is unvoiced, while it would probably be voiced when speaking in a Korean accent.
Bottom line, however, is that the modern romanisation did not change the “p” from McCune-Reischauer because the sound was incorrect, they changed it because they needed a system without accents, using exclusively letters from the Latin alphabet.
Next to this, in “baduk”, whenever I’ve heard it being pronounced by a Korean, it sounds like an unvoiced “p” to my ear (trained mostly by Dutch language, but therefore able to recognise the difference between a voiced and unvoiced consonant). Which would be in line with the pronunciation guides I’ve seen.