Your comments reminded me of my struggles to comprehend the guitar (12-string no less). After many years I finally realized that if you pay attention to the "intervals" between notes - the guitar fretboard becomes almost trivial.
For example:
The (standard) tuning of the guitar is EADGBE. If you look at the intervals (half-steps) between strings, you're looking at 55545.
If you combine that with knowing the intervals between the notes of a chord or melody, it's easy to "do the math" and find your way around the entire fretboard.
For a major chord composed of a root, major 3rd, and perfect 5th (0, 4, and 7 halfsteps): start with the root on the lowest string. The major 3rd is 1-fret down (+5 -1) on the next string. The perfect 5th is 2-frets below that on the third string (+10 -3).
Maybe not the preffered example or fingering, but it's easy to visualize all the various intervals and inversions that can be found with some simple math.
Unfortunately, this system isn't quite as straight forward on the concertina. But I'm learning!!!