I'm building a simple ground plane antenna for VHF, and I'm doing some research on grounding in case I want to permanently install it. I keep hearing that the ground rod is connected to the outer coax conductor, but wouldn't it make sense for the highest (thus most likely to get struck?) part of the antenna to be grounded?
The coax shield must be at ground potential if you want the coax to function as a non-radiating transmission line. If the shield isn't at ground potential, then there will be a non-zero electromagnetic field between the coax and its surroundings. Meaning, it will radiate.
The shield could be left floating, but the shield can be grounded without changing the operation of the circuit since no current will flow in this ground connection since both the shield and the ground are at the same potential. The same is not true of the center conductor.
Furthermore, some antenna types don't give you much choice about which part to ground. For example, a vertical has radials buried (or very near to it) in soil: they are grounded. If you ground the vertical part also, effectively the transmitter is trying to drive a short. That won't radiate very well.
In any case, the current from a strike on the center conductor (or part of the antenna attached to it) is largely the same as a strike on the shield. A good part of the strike's energy is at high frequencies, and thus the current will migrate to the shield due to skin effect, and probably also arcing, either intentional or not. The shield will (hopefully) be effectively grounded, so a good portion of the strike current goes directly to ground. What residual current remains on the center conductor is shunted to the shield by a gas discharge tube or other surge protection device.
A part of the antenna system (system = antenna + transmission line) that is truly grounded can't radiate. Most antenna systems are more efficient when the antenna, the part that is designed to radiate, is up high. So the antenna is deliberately ungrounded. Antennas are usually designed with the assumption that the transmission line doesn't radiate much; if the transmission line isn't supposed to radiate and it does, then that will affect the antenna's radiation pattern, usually not in a good way. The coax is grounded in part to discourage it from radiating. (One other thing is usually necessary to keep the coax from radiating, and that is something to discourage RF from flowing through the shield of the coax. The most practical way to do that is usually to use a balun that presents a resistance to common-mode currents, which flow through the shield.)
Please keep in mind that grounding at RF frequencies is very different from grounding at DC, or at AC frequencies (50 or 60 Hz). At RF frequencies, a connection to ground is only a ground if the connection is short, a small fraction of a wavelength long. If the connection is longer, then it becomes a transmission line that radiates and transforms impedances. A would-be ground connection to the highest part of a transmission line is likely to be longer than a small fraction of a wavelength, so the would-be ground connection will just be another wire that is part of the antenna and radiates.
While I'm on the subject I'd like to mention lightning, even though it's not directly relevant to your question. Grounding against lightning is different from any other kind of grounding, because lightning has plenty of power to arc through air and burn holes through insulators. The idea there is to dissipate the power of the bolt into the earth as quickly and efficiently as possible, to reduce the damage radius.