Anyone know why all the knobs, light switch, radio knobs, turn signal, etc. become sticky?

Jelly fingered kids?😕

On one of the TV car restoration shows last year, someone had an expensive sports car with the same issue.  I believe they cleaned the knobs and repainted them.

Age, sunlight, heat, skin oils, etc. They all contribute to the deterioration of plastics. Sometimes you can clean the stickiness off but if it's been there for some time, it's probably penetrated below the surface layer.

👆 What he said. The steering wheel on my '74 got to be unbearably sticky. I used 3M Adhesive remover to wipe it clean, and it worked great, with no damage to the wheel at all. Any "soft" plastics like this will eventually get sticky, as the chemicals used to form them ooze out over time. I have a Corvette refrigerator magnet on my toolbox that is sticky like that, Has been for years.