The thing Tsubone did, was actually really clever:
Alluka can only grant a wish, after 3 of her requests were fulfilled.
And (as you said in the video) she will ask requests of only one person until the person has either fulfilled 3 requests or failed to fulfill 4 requests.
Since Killua already made a wish (kill my mother or kiss my check depending on wether we're down the mountain in half an hour or not), Alluka's "request counter" was set to 0.
This means Killua can now only make a wish if Tsubone "allows" it by fulfilling 2 more of Alluka's requests.
So Tsubone deliberately started to interact with Alluka as soon as she met her in order to heighten the chances that Alluka would make a request of her.