Enough of this.

I agree and disagree.

Did the spell cause such a high-speed movement? Absolutely. No question there.

What I'm not understanding is how this idea came that Killua can't replicate such movement on his own. So the next obvious question is: Why didn't Killua move like that in the first place? The answer should be obvious, it's because Illumi was messing with his will to fight back right from the get-go, as soon as Shoot closed the distance that first time Illumi was already in his head. Don't forget that the needle also indirectly slowed his reactions when trying to fight back, something Shoot himself noticed - and then when Killua got in actual trouble, the spell took over forcing Killua to do what Illumi wanted, not what Killua wanted.

See the difference?

For a more direct chain of events, let me lay out how I interpreted it

Killua wants to fight back ---> Illumi's whispers and influence won't let him, he becomes confused, hurt and scared and thus it stunts his overall output. He's freaking out. He tries to have a go, he says himself that he has to do better than just dodge, but as soon as that pulsing reminder of his heritage comes up he second guesses himself and he freezes up.

Killua gets in trouble ---> Illumi takes over and Killua's body escapes with the speed Killua can obviously already use, but not to attack, to escape. Because that's what Illumi wants.

This to me makes much more sense than some needle-boost.

Also, when Killua was freaking out about how Shoot is so fast and strong, that he's better than him he was already under the needle's influence. He wasn't in the right mind-state to make calm deductions, he was freaking out indirectly to the needle's influence.