"But current kills, not voltage"
That's a vast oversimplification. For the same resistance, more voltage results in more current.
Static discharge is only safe because there's very little energy involved - it lasts a very short time and there's not a lot of stored energy behind it, but there's still a lot of current flowing. Take an old TV with a CRT, turn it on, then unplug it. Now try to remove the anode connection to the CRT by hand (not to be tried by those who value their health). The voltage is similar to that of a static spark (around 25,000 volts), but you'll get a really good jolt out of it. There's much more stored energy behind it.
Compare that to grabbing the posts on your car battery, which can put out hundreds of amps for relatively long times. Nothing happens, because of the low voltage. Now compare to getting hit with lightning.
Voltage can be dangerous.