Some of my students are the same way, and I certainly appreciate appropriately applied wit, but often what they lack is timing. It's such an essential part of comedy. Not just knowing when to make a comment, but when not to. Knowing when to wait a beat before raising an eyebrow or giving a look of shock.
The best comedians have the timing down. Bob Newhart, in particular, is a master. He can take simple phrases and turn them to comic gold all through his delivery and timing.
I don't claim to be anything near that level, and sometimes my timing is terribly off. The worst events are when I misread a situation entirely and try to insert a joke or humorous expression when the timing is entirely wrong. Thankfully, those moments are getting fewer, although I still run into that problem when I am too desperate to be funny (which is also, thankfully, a declining occurrence).
I suppose timing is important in aspects of our lives other than humor. Maybe that's why comedic timing hits such a chord: so much of the timing in our lives is out of our control that it's amusing to see a person take control of timing and use it deliberately to her or his advantage. Timing might not be everything, but it certainly counts for a lot.