There probably is a better or more general notation system, but specifically for western music it is actually pretty efficient and logical once you start working with it a bit. Just don't put too much weight on the names and think in intervals. You have a scale made up of 7 notes/intervals with the "fifth" simply being the fifth note in the scale. Same with third, etc.
The specific flavor (major/minor) of e.g. the third you're playing usually depends on the mode, but it is still a "third" and serves the same-ish function.
Extending the names i think would actually be more confusing. I'd argue it already puts the patterns of scales and chords in the foreground.