Db would be 01. Dbb would be 00. (or 11 and 10 for the octave)

The point ajuc is making is that the flat-sharp accidentals aren't used or needed at all if you just assign numbers to each tone. There's no concept of flat or sharp, unless you want to deal with microtones.

The letters and flats/sharps give you key/value over frequencies, which is better than just a numeric index over a chromatic scale. Working with the keys allows for the same abstractions to be used with all 12 keys at the same time, on the same staff.

I'm not really arguing one way or the other, just pointing out that they're functionally equivalent. It's two different maps keying to the same set of frequencies. The BASE12 system described above would allow for keys as well, just with a different notation. Instead of flats/sharps marked next to the clef in traditional sheet music, the BASE12 system could start each line with a list of 'prohibited' notes. For example, when indicating that a piece is in the key of Gm, the staff in BASE12 could start with : [01,04,06,08,0b]. This indicates that the majority of the song will be made of the notes 7,9,b,0,2,3, and 5, and serves the same purpose as having two flats next to the treble clef, one on the middle line and one in the top space. Which of these systems would be easier for humans to grok is up for debate. I personally don't think either one is better.

Thinking about it, the traditional notation is just mapping to an octal system, with the key accidentals acting as modifiers to the map and the base-8 values being displayed graphically as vertical position on the staff.

You need to stop thinking about things in engineer terms. The scale going 7,9,b,0,2,3 instead of G,A,B,C,D,E,F with B and E flat may be functionally identical, but one is sure a better representation of the actual patterns in the music than simply referring to a number proportional to the log(frequency).