To put it another way: The thing to realize is that both Gatekeeper and the Exile mat have when-gain abilities, so they trigger at the same time. You can then choose which to resolve first: Exile the card for Gatekeeper, or discard copies of it from the Exile Mat. If you first Exile the card and then discard Exiled copies of it, you would discard that very card unless it said "other".
Note that there are several other cards with when-gain abilites, which you could choose to resolve before Gatekeeper: Cargo Ship, Changeling, Innovation, Royal Seal, Tracker, Travelling Fair, Villa, Watchtower and Inn. All of these would work as hard counters to Gatekeeper. (See the rules document in my sig, p. 31, for more on this stuff.)