All the above is correct, but you need to consider a few other things:
You do need to have all your grounds bonded per NEC.
You also need to consider lightning grounds. A lightning strike won't take a wandering path to ground. Some of the current will follow this path, but the majority of it will look for a more direct path. That could still cause issues if you take a direct hit.
Ideally, you'd want a ground lead coming straight down from the antenna mount to at least one ground rod. Ideally no bends in the down lead. You'd want your coax shield grounded, also. Installing a PolyPhaser type lightning protection device on the coax before it enters the house would be common practice. If the path from antenna to PolyPhaser is short, the PolyPhaser will ground the shield. If it's a long run, common industry practice to to ground the coax shield near the antenna and again near the bottom of the support structure. If it's a -really- long run, as in tall tower, you'll often see the coax shield grounded a several places along the length of the tower.
Being able to give a lightning strike a direct path to ground is important. Bonding all your grounds is important (and required). You also need to consider the nearby strikes that can induce a lot of energy into your antenna system. The PolyPhaser type products will help address that.