No airport in the US in their right mind is going to allow un-docked scooters to be left lying around their property.
Look at it from the TSA security point of view - dropping a scooter off is tantamount to leaving a suspicious package (full of wires and batteries to boot) lying in the middle of a vulnerable location. And even if you had docked scooters, there is no way to validate that a scooter brought into the airport has not been tampered with. So again that won't fly.
The TSA is very nervous when it comes to security, so anything that could even appear to compromise that stance is not allowed (even if those policies are full on cray-cray at times).
That being said, an airport such as LAX is not really a suitable location for scooters in the first place (and I know LAX from many trips there). You have thousands of people in a rush to get from ground transportation to gates or from gate to gate, and a large number carrying large amounts of luggage with them. And did I mentions the stairs? None of that would suit itty bitty scooters which nominally only support riders under 200lbs (and the average weight of adults in the US is 180lbs - but I swear that there must be a lot of skinny people in the US hiding in the woodwork)
So forget the scooter and do what everyone else in LA seems to do - use a car or a bus.