Notes in different octaves are different and hence should look different. Even if you restrict to keyboard instruments, I am not sure you describe an existing problem with classical notation.
For the non-keyboard instruments I know how to play, tones in different octaves having different identifications is a must unless you want to add extra parts to the notation.
I think the point is that the standard clefs would be better aligned if there were 6 lines instead of 5, so both bass and treble could be EGBDFA. It would put the low C one below the bass clef and the high C one above the treble clef, instead of two below and two above but only one in between.
Yes, that was my general thought -- (added:) but that still would have two octaves different, where a note between staff lines in one octave is on a staff line in the other.
Also when you go above or below the two clefs and draw ledger lines for those notes, you're losing the background pattern once again -- you start having to count notes. With this proposed notation you'd draw another background octave (or enough of it for the notes used).
For the non-keyboard instruments I know how to play, tones in different octaves having different identifications is a must unless you want to add extra parts to the notation.