I don't know too much about this, so I definitely could be mistaken, but the professor seems to say this hieroglyphic may be connected with our current letter "B". Also, I think he said something about "house" on another video and I believe it was different.
When it comes to that video, surely the professor would be aware that lowercase letters are a late invention, b is just a version of the original B, so the lowercase shape per se is irrelevant. Furthermore, I don't think there is any doubt in anyone's mind that B comes from semitic beth, so whatever you do, you need to compare THAT shape to Hieroglyphs.
𐤁 in Phoenician, ב in Hebrew.
Check out https://en.wikipedia.org/wiki/Bet_(letter) and scroll down to the comparison between Egyptian and the Semitic languages.
And again, there's also the name of the letter, which means house, not leg.
The Wikipedia article for "b" seems to list both the "foot" and the "house" hieroglyphs in the development on the sidebar... not that Wikipedia is always correct.
I think the professor in that video is quite well known and has written several books on Egyptology, so I doubt he would be completely off base on something so basic. In any event, you're absolutely right that there the house hieroglyphs is related. I was wrong to question that. But it seems that the "foot" hieroglyph maybe has some relation too. If you find anything more about this, I'd be curious to hear what it is.
"The Egyptian hieroglyph for the consonant /b/ had been an image of a foot and calf ⟨ Leg glyph ⟩, but bēt (Phoenician for "house") was a modified form of a Proto-Sinaitic glyph ⟨ Bet ⟩ probably adapted from the separate hieroglyph Pr ⟨ Per ⟩ meaning "house"."
It doesn't seem from this that they are claiming that B comes from the leg glyph. so it's very strange they put it in the side-bar, if it's etymologically unrelated.
Also, if you look at the Aleph article, https://en.wikipedia.org/wiki/Aleph, they have a sub-section with some Hieroglyphs that are pronounced as aleph, without mentioning any historical connection, which seems very strange. The rest of the article talks about letters that are historically related, so why bring up the vulture glyph? In the article for a they don't show the vulture though, only the ox.
I think part of the reason people say things like this is that they are so strongly inclined to see writing systems as alphabets, so they just have to find "the letter b in Egyptian", even though that makes no sense at all. Egyptian didn't have a letter b, they had a number of glyphs that represented 1, 2 or 3 consonants.
Btw I'm sure the professor knows his Egyptology, so the preceding wasn't referring to him, but maybe he didn't study the origin of the Semitic writing systems. That's not really Egyptology.