I have recently been wondering about what I really want a tweeter to do in a system.
This is true for many. I have grown out of sparkle, and air, and that seems to correlate well with smoother FR and lower distortion (when I've been able to measure them).
That doesn't mean you should. My point here is there's something like what I call a residual sound. Like scented body wash, and in the right music, environment that can be a lot of fun.
The most natural tweets, like Troels Graveson has said, to me, disappear. They sound like they are simply not there. The best AMT's can be there, as can the best diamond, and Be, and there are a lot of soft domes and ring radiators which are not bad in these terms.