I assume your music education
may be limited so please don't take it personally if your question and explanations here were hypothetical.
As for middle C, it's usually the starting point in Piano lessons due to it's simplicity to understand and is in the middle of the keyboard. You are incorrect suggesting that it was made that way for ease and location.
The musical scale was discovered centuries ago by a Greek mathematician and the intervals of semi tones and whole tones have long been determined. So your hypothesis would suggest that a extra black key in every octave would have made middle C harder to find so pianos weren't built with it. That is incorrect.
As for sharps/flats, it is key dependent if a sharp or flat can be a black or white key. The F# Major scale has an E# in it, yet the key one would visualize if they didn't know better is the F key. That would be a enharmonic equivalent. This same scales enharmonic equivalent would be the Gb Major scale which has a Cb in it's key signature yet one would probably view that as a B.
All instruments of pitch follow the same rules yet obviously not all have white and black keys or frets to locate notes. This is where knowing and hearing the difference between whole tones and semi tones come into play.
Here is $20.00 worth of music lessons...
White keys and black keys can be both whole tones and half tones dependent to what key you are playing in. Take any key of the Major scale and the formula in order of whole tones and semi tones are the same, i.e.
Whole Tone - W.Tone - Semitone - W.Tone - W.Tone - W.Tone - Semitone- W.tone (octave)
So yes a C major scale, the most basic because there are no "black" keys is nothing more than C D E F G A B C and the 2 semi tones are between E-F and B-C .
But if you played an E major scale the notes would be E F# G# A B C# D# E and the semi tones would fall between G# and A and D# and E
So you can see the semi tones are always after the third note and the seventh note from the root in every key signature of the Major scale and could be a black key or white.
When you learn the basic building blocks of scales and chords and how they are constructed, the formulas are the same in every key signature.
Music theory isn't hard at all, in the beginning at least. A lot of it is based off formulas and methods to ease the learning process. It is in the grasps of everyone but I'm sorry to say that most people are too lazy and make up every kind of excuse to justify not learning it.
My theory is not limited, but it is not advanced at the same time. I do understand all that you have mentioned, in bold. So that makes up most of the post lol.
I was not suggesting that adding an extra black key was avoided due to the pianos construct. At least not so directly. But more as, the piano was constructed in that fashion so that middle c, any c, any note at all can be located. If, there ... to be continued