It's just model differentiation and Lexus are very inflexible regarding mix and match. They would not entertain any deviation from the options listed in the brochures.
Pulling left and right is often just tramlining, wide tyres following imperceptible ruts in the tarmac cause by trucks/heavy traffic. White lines can also be a problem.
Some tyre makes are worse than others.
Something I've noticed when watching these videos. I know we (the viewer) are expecting something to happen, but come on some are so obvious that one can only think the driver is looking to prove a point, often accelerating into trouble.
Does the car have a sticker giving the kerb weight, gross weight and most importantly in this case a train weight?
If not then the vehicle isn't type approved for towing and one's insurer would take a very dim view if one were to tow and had an accident.