It’s already been pointed out that the static charge is generated when you slide across cloth seats getting out of your vehicle. But the charge isn’t building up on your vehicle, it is building up on you. Because the tires have a compound that makes them conductive, any charge on the vehicle bleeds off to the ground. But with your rubber soled shoes, you are insulated from the ground and the charge bleeds off more slowly to the air. If you then touch anything else that is grounded and conductive, like your car door, the charge bleeds off very quickly and you get shocked.
There is a very simple and effective way to prevent this from happening, and it doesn’t involve dryer sheets or anti-static spray. All you need to do is to maintain contact with a conductive part of your car while you get out of it. The metal door frame works somewhat, but being painted, it isn’t the best option. If you have any bare metal or chromed parts that you can easily keep one hand on, this will work better. The way this works is that by grounding yourself, no static charge can build up on you as it will be drained away while it is being generated and before it can build to a level to deliver a nasty jolt.
I’ve used this method for nearly 20 years and it works every time. My only problem is that newer cars don’t have much in the way of chrome or bare metal parts to hold onto, and now I have to make do with keeping a firm grip on the painted door frame, so I occasionally feel small shocks as the charge builds up enough to overcome the insulation of the layer of paint.
To answer the question of old tires vs. new tires, the only answer must be conductive vs. non-conductive. If your tires are non-conductive, then they will insulate your car from the ground. In this scenario, you get out of the car and acquire a static charge from sliding across your polyester blend seats. You touch your car and some of the charge from you transfers to the car, but since the car is insulated from ground, the charge doesn’t flow to ground and you don’t get a shock, or possibly a minor one.
On the other hand, if the tires are conductive, then the car is always grounded through them. When you get out of your car and build a charge, touching your car allows that charge to flow directly to ground very quickly resulting in you getting zapped.
For this reason, I have serious doubts about the grounding straps discussed on the show having any positive effect on this problem, since the tires are already conductive.