My biggest objection to conventional notation is that it gives a profoundly misleading picture of how music and harmony really work. It defines one reference key (C Major/A Minor) with a certain pattern of steps and gaps, starting on a certain note. Then for all of the other keys you add more and more sharps or flats until you get into ridiculous keys where all 7 notes are modified. The truth is that there's just one evenly spaced set of 12 tones, and all it means to be "in a key" is that you've picked a certain note out of the 12 to start the pattern on. There's nothing special about C. We could have chosen the key we call F# as the reference key and named it C, and everything would work the same.
It's hard to overstate the damage from this. Lots of musicians I know—serious players, people who took music in college—still think of "complicated keys" and "easy keys" and are only vaguely aware that the keys are actually all the same and they're just being tormented by the notation and terminology. I'm teaching guitar to a friend who was first trombone in high school and it blows her mind that she can play the same scales starting anywhere up and down the fretboard and it sounds the same.
It all comes from the design of the keyboard, where the notes of C major are evenly spaced (white keys) and the sharps/flats are stuck in between. There's also the fact that in the past the 12 notes weren't evenly spaced, so the different keys really did all sound different back then.
Conventional notation does have one big advantage, though: every line or space represents one note in the scale. This is more how musicians think: you don't care that much about the notes outside your key, and having the other ones "tucked away" in between makes it easy to see what's going on. That's why it's so quick to read once you know it. Out of the hundreds of alternative notations, I haven't seen one that's both key-neutral and also makes it easy to see things in terms of scale degrees.
(One idea I've had is a 12-tone staff with Sacred Harp-style shaped note heads to show you what scale degree you're playing. Not sure if that's ever been tried.)