In every world except Garfield's, automatic doors have sensors to protect people from being smashed in their mighty jaws. In Jon's double-barreled dumb-move, he not only managed to injure himself in an impossible way, but wants to twist the focus into a joke about our horrible relationship with technology. It's not that joke. It's a story about Jon's good intentions being squashed under the weight of his zealousness. It's about unnecessary, overwhelming desire to please others, and being thwarted by your own stupidity and inabilities. Jon feels on some sublimated level that his chivalrous intentions are a positive trait, and refuses to acknowledge that the automatic doors of the Garfield world are telling him otherwise.
Garfield, meanwhile, has concocted a coping strategy that leaves his arm unwrenched and ego unbruised: he doesn't try to impress anyone, and pretends he doesn't care about them.