Why are the notes named what they are "instead of something else?"
All note-naming schemes are ultimately arbitrary. There are, in fact, many other systems for naming notes. For example, a good portion of Europeans use H for B and B for B♭ whereas, in other parts of Europe, the notes are named Do Re Me Fa Sol La and Ti (or Si). Sa Re Ga Ma Pa Dha and Ni are the note names in Indian music. All these systems evolved over centuries.
Any new and potentially more logical system of notation would be equally arbitrary. For example, MIDI, which was developed in the early 1980s, uses 128 different note "names", which are really bit strings and can be represented by decimal numbers. Middle C, for example, equates to binary 111100 or decimal 60.
You asserted, "it would be possible to include [systems] where all symbols are letters or that the extra 5 notes without letters are after B." That's true. You could devise a system where the notes would be:
- A (same as C)
- B (same as C♯ / D♭)
- C (same as D)
- D (same as D♯ / E♭)
- E (same as E)
- F (same as F)
- G (same as F♯ / G♭)
- H (same as G)
- I (same as G♯ / A♭)
- J (same as A)
- K (same as A♯ / B♭)
- L (same as B)
This may have some advantages (and a new set of disadvantages), but you can't expect it to replace the existing systems any time soon.