Continuing:
What he does is similar to how I used to "read" music for decades because I had no training whatsoever and had to find my own way. I had a kind of reference line, which was the Tonic, and saw (and heard in my head) the notes as they went up and down from that reference line.. I related to music in relative pitch, and everything was highly diatonic. The major Tonic was Do, the minor Tonic was La. The piano likewise had a reference spot. If in D major, the "reference spot" corresponding to the reference line was the "D key" on the piano. This works especially well for melodic passages, or music with interweaving melodies, or something with simple predictable accompaniment. I played mostly Clementi sonatas back then with its ubiquitous Alberti bass which is both (because that's what I had for playing.)
For the transposing question. All I had to do was pick a different piano key as my Tonic while still eyeballing the original score for its Tonic, and go "above and below" intervallically the same way. In fact, back before I knew much, I did that accidentally, and wouldn't know I was in the wrong key until I had to hunt up a bunch of black keys to make it sound right. Had I also learned to play scales in different keys, and how keys worked, it would have been different. I could do that now. You asked why I "don't do that anymore". Who said I didn't? From time to time I pop over to that old world and explore.
The "key of C major" also represents diatonic notes of music in rather ordinary major and minor, in a very visible form, i.e. "all white keys". My mental soundscape, since I thought in major and natural minor, did the same thing. If a piece is in E major, and very diatonic, then you have your 4 sharps: You can draw an imaginary coloured line on "E" in the score, and maybe another on B - and if you do that, you could easily transpose that music on the fly to any key.
Supposing that song is Twinkle: E E B B C# C# B ........... and I decide to "play it in Db major". How I'm actually perceiving this in the music is: Tonic - up 5 (P5), up one (m2), back down (continues down the scale to the tonic for "how I wonder what you are". So "Db" is "that black key below it, and then "up 5" etc. If I have a solid handle of the Db major scale and thus Db major (visibly on the piano it maps out as all black keys plus two whites) then I'm good to go.
Essentially, that is how I transposed using my old frames of reference. The lines and spaces sort of "erased themselves" except for the "reference line (or space". In fact, there are some old scores where I literally drew in that line using a highlighter.
Since I primarily hear music in relative pitch, and only recently learned to perceived "A as A", transposing this way is relatively easy. Here, "perfect pitch", i.e. hearing pitches as pitches, is a handicap.
To bring this round to the alternative notation; what he is doing seems similar to how I used to relate to music, except that he uses middle C as a reference line. I don't like that much, and I also don't want to perceive music in relationship to C, or middle C. There are a few reasons for that.
I also noted the idea of changing stems to indicate octave below or above in his notation. Piano music spans a LOT of octaves, and some music jumps around tons, in wide leaps.
What I do see here is that it may give other ways of perceiving music - maybe some angles that some of us already have (or not). Perceiving music via multiple angles is very useful.
Meanwhile, the person who created the notation and is demonstrating it has his own musical background and will be drawing on all of it. Some may be subconscious, even to himself. Whatever he demonstrates will come from his abilities and background, and not just the notation. If you yourself find it expands what you can do, and/or how you perceive or work with music, well that's fantastic.
(As an aside: That cup of coffee or tea on sitting on the keyboard makes me cringe.

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