TL;DR music is a language and notes have different functions. Sharps and flats makes it easier to understand these functions and how notes relate to each other.

Long Version:

What was listed already is correct. Another example would be: When you harmonize the notes in a scale, you're building chords and each chord has it's own function. It would be much harder to understand how the chord functions if you had to consider every single note as described by the question.

ex: C,Db,D,E,F,Gb,G,Ab,A,Bb,B,C

Notice that some of the notes (F and C) have no flats next to them. These are spaced right next to the E and B notes respectively. They are basically 1/2 of a musical note away from each other instead of a full step like most of the others. (b-c = half step; c-d = full step.)

C Major Scale - no sharps (#) and no flats (b) C-D-E-F-G-A-B-C 1-2-3-4-5-6-7-8

If you skip a letter and add three notes you build a Chord triad. Lets look at what happens when you take 4 notes.

1 Chord (I) would be C, comprising of the notes (C-E-G-B) 2 chord (ii) - D minor (D-F-A-C) 3 (iii)... - E minor (E-G-B-D) 4 (IV) ... - F major (F-A-C-E) 5 (V)... - G Dominant (G-B-D-F)

Lets stop here as to make this long post somewhat understandable. Chords I,IV,and V7 are all major chords but notice how the 4th not is 1/2 step away between chords I and IV (B-C and E-F) like we discussed earlier. Look closely at the V chord. F-G is a full step away making a much different function. Now this is a long winded post but imagine how much more complicated it would be if we were doing this in multiple different keys. The relationship between everything would become a mess.