I think you've put your finger on a dubious feature of the KCA. While I would say that a chair does begin to exist when it's created from pre-existing materials, I agree with you that if the universe began to exist, the universe didn't begin to exist in the same sense in which a chair does.
So I think you're right to detect a questionable move from "Everything within the universe that begins to exist has a cause" to "Everything, including the universe itself, that begins to exist has a cause." It's not at all clear that the phrase "begins to exist" is being used in the same way both times.
To the question "You've never seen a chair pop into existence, have you?" one can reply as follows: "I've never seen anything arising from pre-existing materials pop into existence, but that isn't relevant to whether something not arising from pre-existing materials can pop into existence."