The purpose of the dome shape is rigidity. Domes are very strong for their weight.
At frequencies where the dome is effectively rigid, the dispersion is essentially that of a piston of the same diameter. The dome shape does not improve dispersion in and of itself because it's not a pulsating dome, it's a moving-back-and-forth dome.
At frequencies where the dome is in breakup mode, the dispersion tends towards that of a ring radiator, with the diameter of the ring being the diameter of the voice coil. Some of the inverted dome tweeters have a voice coil that is smaller in diameter than the perimeter of the dome, and therefore have a little bit better dispersion when they go into breakup mode.
A pushed-in tweeter dome no longer has the geometric rigidity it was intended to have, and even if you manage to pop it back into shape, the material may have been weaked where it was creased and so its behavior may be degraded.
Duke
dealer/manufacturer/occasional user of dome tweeters