If you start with C major (no sharps), and ascend in 5ths (G, D, A, etc.) then you add one sharp to the previous major scale to get the major scale starting on the new note. The added sharp each time is the 7th of the scale. So G major just has F#, D major adds also C#, A major adds also G# etc.
If you start with C major (no flats), and descend in 5ths (equivalently: ascend in 4ths) (F, Bb, Eb, etc.) then you add one flat to the previous major scale to get the major scale starting on the new note. The added flat each time is the 4th of the scale. So F major just has Bb, Bb major adds also Eb, Eb major adds also Ab etc.
Key signatures put the sharps or flats in this order.
The two meet in the middle:
F# major: F#, G#, A#, B, C#, D#, E# (key with 6 sharps)
Gb major: Gb, Ab, Bb, Cb, Db, Eb, F (key with 6 flats)
I.e. on the piano F# and Gb are the same key, G# and Ab are the same key etc. H J K A C E F
First of all, which key is it? We know it starts with H. But is it major or minor? We have to mentally calculate all the intervals, rather than just looking for the sharps and flats.And calculating the intervals is annoying! You could go 2122212, which doesn’t correspond to major or minor — it’s dorian.
Meanwhile, in current notation, you’d just look at the sharps (C# and F# = D major) and the starting note (E).
I think in an alternate universe where a 12 (or 6+6) notation had historically been used instead of 7+5, the 7+5 option would seem quite unnatural to most people. In any case, starting from a blank slate without historic context, the 7+5 naming scheme just isn't intuitive.
Anyway, you can count the intervals now. For example, I know that there are five half steps between F and A#. It is slightly annoying to keep the B and E exceptions in mind, but it’s definitely possible — most people just find it easier to use the sharps and flats as a shortcut.
A 6+6 system makes more sense in terms of fundamentals - there's the exact same distance between each "full" letter. That doesn't limit what kind of scales you can annotate with the letters at all.
C D E E# F# A# B#
In fact, I think every diatonic scale in 6+6 notation will have three or four sharps/flats.