No, the statement is not false. You are confusing this statement with the arcing. The only reason why I chose an example that showed arcing is because it was a
visual cue to you and others that current was flowing.
When the technician touches the line with his electrode, current is still flowing through the electrode in the same manner as when it was arcing, but the results are not as dramatic and visible without an ammeter. (By the way, notice that his electrode still arcs when he releases the helicopter to move away.)
The reason why these guys wear a full-body Faraday suit is to prevent a voltage gradient from one part of their body to the next. If we were dealing with lower voltages, this suit would not be required, because the entire body would be able to maintain the same voltage--like a bird landing on a power line. However, you will not find a bird resting on one of these ultra-high voltage lines, because there will be a voltage gradient across the body of the bird and they will be electrocuted. Without the Faraday suit, the technician too would be electrocuted, simply from the voltage difference from one part of his body in contact with the line to another part of his body away from the line.
We all know that under normal circumstances we can grab a live conductor and not get zapped as long as we are not in contact with another voltage potential. However, when you are dealing with these ultra-high voltages, the minor difference in distance from the line will create a voltage gradient. If you touch one of these lines, you will be electrocuted from the inside-out. That is to say, you will have a current flowing inside your body, regardless whether any current is entering or leaving your body from the actual contact.
So yes, current will flow any time you have a difference in voltage. If you still don't believe this, I will pay you $100 to climb the power pole and get within a couple feet of the line without a Faraday suit. (I'll make the check payable to your widow, of course.)
Disclaimer: I am joking!! Please don't do this!!