Today I took my Aimpoint off and did some point shooting.  For 10yd targets, instead of anchoring my nose to the charging handle I looked over the rear sight and put the front sight just below where I wanted the bullet to impact.  It was faster compared to finding the dot or lining up the sights.  Double taps were a breeze.
Still, at medium ranges the dot worked faster than lining up the sights, but the body of the Aimpoint got in the way of point shooting.

I've never practiced point shooting before.  Is that how it's suppose to be done?

Thats not "point shooting".  What you describe is still aimed fire.  Massad Ayobe (Sp?) calls it "stress fire". For real and true "point shooting" the gun should be below your line of sight so you can not visually index of any part of it.  Of course opinions will very ...

Try an EOTech. I think they are a bit faster than Aimpoints or ACOGs

use the body or the aimpoint as a giant aperature.

I'm perfectly fine with my Aimpoint.  As said above, use the body as an aperature.  If shooting with both eyes open, you should see nothing but a red dot, as if it were a laser.  Maybe a slight haze of the outline of the Aimpoint