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.