I think that the role of counter-intelligence is most of the time not really catch anyone, but to 1) get information from an adversary, and 2) develop methods which allow to protect own information against adversaries.
But I've always understood a part of counter-intelligence is to counter the intelligence gathering of an adversary, or I suppose ally as well depending on circumstances. Which is what you are saying with your second point. Therefore, there is someone to catch doing it, just in a broad definition of the term.
