Speaker cone shape


Why are speakers cone shaped, apart from rigidity? To my mind the air being pushed by a cone would radiate at an angle inward toward the axis of the speaker and collide in the centre, which seems inefficient to me, and likely to cause some distortion of the sound. This may also cause interference to adjacent speakers on the same baffle.  Would there be any advantage to having the surface flat, assuming you could maintain rigidity without increasing the mass? There must be modern capable materials out there.
Is the fact that the speaker is cone shaped that causes the volume to change counter intuitively as you move left and right in front of the speakers? What I mean by counter intuitively is when you move left the right speaker sounds louder and visa versa.
chris_w_uk
But as the surface is at an angle the air can’t be pushed along the axis of the speaker only toward the centre. If you move something through the air at an angle the air doesn’t get pushed directly forward, it always moves in the direction of the trailing edge, which would be the centre of the cone.
There's all kinds of resonant modes in a given cone shape. If it's a standard typeish round cone, you can pretty well tell which modes it's experiencing when looking at the FFT plots/graphs with an educated eye.

This is a fairly deep subject. When exploring it, it is easy to hit the weeds. Meaning... lots of arguments, discussions, and complex nebulousness to be explored.
They're not all shaped as cones. There have been a number of flat drivers, and one current example is the top model from Vienna Acoustics. 
I just had a look at Vienna Acoustics, even those are not completely flat, but very nearly, well that's how they come across in the images.

They say: "The plane surface of the spidercone offers the advantage of precise, piston-like movement. Any diaphragm should move in and out as one unit, not as a quivering mass of uncorrelated vibrations. It should move as a perfectly-solid piston. The flat cone is extremely stiff, with bracings of a height of 18 mm (bass) and 14 mm (mid), yet light, respectively, further reinforced by glass fibers in our new X4P mixture."