Grid layout isn't about borders. The pieces could be shaped like Escher-esque duck tesselations for all borders matter. The border still ends up surrounding regions of control. The shape of the constituent pieces is unimportant.
The critical issue is that the grid dictates unit movement, which is clunky when working with squares. If you want to go "one square" northwest, you have to actually move one north and one west. With hexes, you just move one hex NW. Unit movement behavior. is more consistent and dependable that way.
Civ V draws its borders with some consideration for what you're getting at. They're not harshly hexagonal. More curved and smoothed.
And, anyway, the earth is a sphere. Round makes more sense. Sure, western US states might be square, but pull up a world map. Most borders are anything BUT square and pretty lacking in right angles.
Here's an example of how Civ V does borders:
On the subject of the game itself, I'm having a blast with it. The new features change the early game, and for the better. One of the best improvements is the ability for cities to defend enemies at their borders. Each turn, any city with enemies in range (a couple tiles) can launch a barrage. It makes barbarians and wandering armies much less of a coin-toss game-changer in the early hundred turns or so.
City states are also a neat addition, providing new income and unit generation possibilities. And it's pretty satisfying to come to the rescue of a little city-state who is under siege by barbarians.