Algiers was officially at peace with UK.
This allowed Lord Exmouth to cross the most dangerous part of the approach, where algerian batteries would cross fire and use the longer range of their heavy guns.
He could then anchor his fleet at leisure (a dfficult operation if fighting, the current is strong and oriented to the shore at Algiers) at gun-range distance of the walls, then sent an ultimatum, threatening bombardment if the Dey did not comply to it.
That the first shoot was algerian is then anecdotical.
And you are right, no loot there, just another "pre-emptive war" - with few long-term result : UK had to send another expedition a few years later, and the question will be settled only by the french conquest of 1830.