High voltage is dangerous for the reasons you say, it's just that 12v isn't remotely high.

You don't see warning signs about current because the current passing through the wire is irrelevant - it's the current which will pass through you that's the problem. Which as we said is a lot lower as the resistance of a wire is tiny compared to the human body. Voltage is measurable and consistent - 10,000 volts is always 10,000 volts - the current depends on what it's passing through. "Will cause dangerous amperage when applied to the human body" doesn't have the same ring as "high voltage".

Voltage is the "push" and current is the end result of that push. Imagine a tube full of ping pong balls - you apply a force to one end and the balls start moving through the tube. Fill the tube with Vaseline and apply exactly the same force and the balls coming out the other end will appear a lot slower. This, incidentally is a good analogy to understand why electricity moves extremely slowly but its effect moves at nearly the speed of light.

Anyhoo "current kills, not the voltage" is a terrible statement but has the right intentions. It's like saying "it's not the bhp of the engine that kills you, it's the speed of the car"... Strictly true but one drives the other. Electric current kills you for the same reason it does everything we want it to - it's what creates a change of state in energy.. To heat, light, sound etc. Bad things when passing through your heart! Most lethalities are with AC because of muscle contractions, again very bad when important stuff like our heart and lungs are muscles.

But yeah, voltage will kill you by driving a current. Current can only exist to kill you if there's adequate voltage to drive it.