For example, if you number the tones and have a piano style instrument with alternating white and black keys, then you can have a 6 line stave where spaces are always white keys and black keys are always lines.
When sight reading chords pianists, for example, don't really think about every note in the chord.
Instead if you recognise the intervals between the tones and the root you can just place your hand without thinking about it too much.
Since most of the time you're in a single musical key you don't need see every tone separately. You can still flatten, sharpen or normal and you can represent key changes too.
There are other details such as beginners often already using numbers to indicate preferred fingering. I guess that's not a big thing but if you use numbers as part of an alternative system you have to cover all of the edge cases and extensions.
Oh, just remembered another reason. Having a more compact staves means you can represent more musical parts on the same page which is important in orchestral situations where you might want to see what others are doing as you play.
Many people have tried to replace standard notation, over several hundred years, and no one has yet come up with a compelling alternative. So it's probably a very difficult task. Systems that are evolved over a long time, like musical notation, are often far more robust than those that are designed from scratch.
If you're really serious about proposing an alternative I would probably start by considering why other competing systems have failed.
I'm not really serious about replacing it, more like writing your own OS: it's fun to do and you get a better understanding of why the existing systems work.