Considering only equally tempered tuning, do full notes such as C, D, E, etc have any special characteristics compared to flats and sharps, so that they 'deserve' a separate letter?
If no, would it make sense to simplify the notation to N1..N12 in case no other tunings except 12-TET existed in the world?
Letter names for notes are a holdover from historical naming conventions, but they are also very useful in terms of representing concepts from tonal music. For instance, the seven letter names without accidental alterations [A, B, C, D, E, F, G] provide a diatonic collection, specifically the C-major diatonic collection. By changing the B to Bb (Bflat) or by changing the F to F# (Fsharp), you still obtain diatonic collections, the F-major and G-major diatonic collections respectively. (See the circle of fifths.) In this sense the notes without flat or sharp alterations are not in any way preferable to those with alterations, they are just different and provide alternative scales.
Historically, those seven lettered notes without alterations were doing exactly what you propose, but they were ordering the notes of a seven-note diatonic scale rather than a twelve-note chromatic one. Your proposal to number each of the twelve notes in twelve-tone equal temperament was finally proposed in the twentieth century after the tuning and usage of these twelve notes as individual entities was solidified throughout the common practice period. Musical set theory often numbers the notes (called pitch classes) as the residues mod 12, specifically the numbers 0–11. This is exactly what your proposal does except that in terms of the modulus 12, the numbers 12 and 0 are equivalent. You can imagine the notes as elements on a traditional clock face. The most common version of this integer notation places C=0, C#=1, D=2,..., B=11. Note that all enharmonically equivalent notes (that is notes that sound the same but are spelled differently, e.g. C# and Db) share a single integer (1 in the case of C# and Db). This naming convention is very useful for modern or atonal applications, but it is less helpful in terms of tonal music, which is primarily structured around seven-note diatonic scales.
One of the main characteristics, on pianos and other keyboard instruments, is that basically the simple letter names are the white keys. Yes, occasionally white keys are called # or b, but that's not the general state of affairs.
I guess on most other instruments, the letter names are more academic. And where they are on the stave isn't too important - C and C# both share the same line/space. But that then poses a problem for your idea. How would your now potentially 12 different notes be portrayed?
