This is a common sentence pattern, but I met several variants that I am not sure if they expressing the same meaning. As far as I knew, too adj to do something means they dont want to do this thing, but too adj just to do something means they want to do. But when I was meeting this one you are never too cool to learn new things, I got lost.
Firstly, I think your second note isn't quite accurate.
John is too funny to go to comedy shows. This implies that since John himself is funny, he doesn't go to comedy shows.
John is too funny to just go to comedy shows. This implies that because John is funny, when he goes to a comedy show he maybe gets involved (heckling the comedian, making side jokes, something like that). For this example to work, the emphasis would have to be on go.
John is too athletic to just go to the gym. Again it implies something more than just going to the gym. He shows off, or works harder than most, or something else other than the simple act of being at the gym.
In your final example, you are never too ____ is usually an argument for doing something. In this case, you are never too cool to learn new things is implying that learning is cool, not some boring thing that cool kids don't do. You are never too fit to run a mile might imply that no matter how good of shape you're in, even if you run marathons, sometimes just running a mile is a good workout.