No, he is claiming that l-CLIQUE requires exponential circuit size (this is the reason that most of the proof talks about circuit complexity and not time bounded Turing Machines). This is not necessarily the same thing as exponential time.
Oh ok, I thought the first post was referring to a single algorithm, not every possible algorithm. My mistake.
Sorry, if I could I would edit it to be more clear. We definitely need to show that no algorithm can exist with poly runtime for all inputs.