When an infant learns to walk, you can often see him kind of leering down at his feet, thinking, "Okay, now I need to move this foot right here, on to that spot on the ground right there." And then he does that, and then he pauses, and says, "Okay, now I need to move this other foot here, on to that other spot right there." And his mind is always on one foot at a time, and knowing exactly where it's going to be placed.
But when an adult walks, he doesn't even care where exactly his feet are going, only that both feet are somewhere under him, and that his weight is distributed over them, and that they are propelling his weight forward without toppling him. So his mind is actually conscious of much more about the relationship between his two feet and how they affect his movement, but he doesn't always know or even much care about which foot he moves or where exactly it lands.
Like learning to walk, modern notation is concerned with the real value of where you are pitchwise at any moment. "Right now I'm singing an A, and now a B-flat, and now a G, and now..." You lose sight of how any one note is actually related to the next. If you get all the pitches right, then the melody will be right.
But like walking, chant notation is ultimately most concerned with moving the phrase forward. It is defined in terms of making the melody. If modern notation is constantly saying, "and now, and now," then chant notation is always saying, "and next, and next." And next I'll be going up a fourth. And next I'll be going down a second. And next I'll sing to pitches right where I'm at."
I like the feeling of knowing that I'm shaping a phrase, not just singing a note. And that's why, even though people may be resistant to neumes, I think it's highly important that people accept that modern notation isn't in the cards for learning to chant. I suspect that if this aspect of chant notation is made clear to people, they'll actually be less resistant to it.
There is a certain strike against your pride in being well educated musically and suddenly finding yourself unknowledgeable about what you're trying to sing, but ultimately I, for one, have found it very rewarding. I just wish I could have seen the reward coming. I hope that people teaching others the neumes make it clear to them why chant notation is important and what they'll gain from it, and how to conceptualize why it works.