I don't think this was the case with this disappearance, because she was seen in LA. I think it is likely she told someone she met of her plans. I suppose it is possible she just wanted to disappear, but I doubt it.
LAX is a 24/7 airport, so nobody is going to grab an adult at the baggage claim area. I could see it happening by a fake ride sharing driver
This is all supposition, since the groomer was scared off, but the thought was he wasn't working alone. You're right, he wouldn't "grab her" there --- his job is to spot the prey, confirm she's alone and no one is meeting her there. He'll ask where she's from, where's she going, where she's staying.
Then he'll phone his accomplices, who are in a car outside the airport, with her description and any other info he was able to get. They might try to lure her into the car (fake uber, maybe) or follow her to her hotel and make a move there.