Disclaimer: This answer was written in response to version 2 of the question, which isn't quite what the asker actually asked.
Is there an historical reason for naming accidentals rather than assigning a letter-name to every pitch?
Yes. The "white notes" came first and the "black notes" were added later.
The letter names of the white notes were established sometime before Guido d'Arezzo, writing in the eleventh century. By his time, the note B had been split into two, square B and round B. He is credited with the invention of the solmization syllables. He was certainly the first to describe them, although the way they relate to the letter names in his system is rather different.
The system comprises three overlapping six-note scales, called hexachords, based on F, C, and G. The C hexachord comprises notes from C through A, so it has neither B in it. The square B belongs to the hexachord based on G (G through E), and the round B to the hexachord based on F (F through D). The former is the basis for the modern sharp and natural signs, while the latter gave rise to the modern flat sign. The square B came to be called H in German.
The German names for major and minor keys, Moll and Dur, also come from this system, because the F hexachord was the "soft" hexachord, the C hexachord "natural," and the G hexachord "hard": mollum, naturale, and durum, respectively.
Each hexachord uses the same six solmization syllables, ut, re, mi, fa, sol, and la. The note A, for example, can therefore be either la, mi, or re, depending on which hexachord is in use. That is determined by the melodic context. C can be ut, sol, or fa. Round B can only be fa, and square B can only be mi.
In Guido's system, there are no other notes. However, the same forces that led to the splitting of B led people to apply chromatic alterations to other notes, such as F#. In such a case, they would call the F# mi and think of it as a note from a fictitious hexachord based on F. These other notes were therefore called musica ficta. This lasted for centuries as musicians continued to expand the Guidonian system beyond its boundaries without abandoning it entirely, so it is not surprising that these notes continue to have a second-class status to this day.
As practice diverged from Guido's system, the twelve-tone system became entrenched with its circle of fifths and its concept of flat and sharp generalized to the point that it can be applied to any note, even one that has already been altered.
The modern seven syllables arose from Guido's six to fit this system, with his ut becoming do, and with the addition of si or ti. This allowed the three-hexachord system to be abandoned in favor of both fixed do, where do is always C, and movable do, where do is the tonic of the major scale or the third degree of the minor scale.
In later times, much more recently, the solmization system was expanded as described in the other answer to allow people to sing melodies using the syllables, but in all of these systems there is at least some vestige of the fact that the white notes were the first ones to arrive at the party.
