The shock isn’t from the electric field, it’s from charge (current) entering your body. You need to be close enough to the wire that electrons can jump from the wire to your body, which pretty much means you have to be touching it, unless the voltage is incredibly high.
It’s the same reason the current travels through the wire in the first place. It can keep going through metal as long as all the metal is touching, but even a small gap in the wires will break the circuit.
That veritasium video is so over simplified that its basically wrong, the electrons moving is what causes the field around the wire and visa versa. High voltage and current AC systems can do this, because the fields that surround the wires are actually significant. Its called Leakage current or Parasitic current.
The field is a magnetic field around the wire, not an electric field. If you have some wire on you, that wire will get a little bit of current flowing through it if you get close.
This is actually how those shoplifting alarms work. Expensive items have a tag on them with a coil of wire connected to a radio emitter. The scanners at the exit of the store create a magnetic field and are listening for that radio wave. When you pass between the scanners, it will cause just enough current to flow in the tag to emit the radio waves and trip the alarm!
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