“I did not score 2 triple centuries, my bat did.” People outside India will not understand this joke. But, anyway, it is like saying, “Height does not kill you, blunt force trauma kills you.” But, a fall from the height caused the blunt force trauma, which killed you.
Similarly, it is true that current kills you, but high voltage can cause the current which kills you.
People talk about the example of a “Van der graaf generator” which produces a very high voltage (>100,000 volts). It does not kill people when they touch it because it cannot sustain that voltage for more than a picosecond. It immediately discharges upon being touched by you. So, you never really experience a 100,000 volts. The voltage comes down. If a sustained 100,000 volts flowed through your body in any sustained manner, you will be fried and people would not even find your bones. They would all turn to ashes.
A live power line can do exactly that. It can provide a sustained voltage, so it can kill you because it can produce a sustained current. Voltage produces current which kills you. So, voltage kills. Just like height kills.