Outside of wins, It has to be goals that don't interfere with gameplay. "partial wins", if you will. Using Bilgewater examples, ARAM towers destroyed: good. That's the only way to win ARAM. Total minions killed: bad. You get people like me sitting in TT farming wave after wave refusing to fight the enemy until they complete the quest.
Come to think of it, there are only 2 objective "partial" win cases in LoL. ARAM towers destroyed, and Dominion nexus health drained. Anything else has the potential to be a bad play. Going for too many towers on TT or when a push could win it, taking TT altars or Vilemaw at the wrong time, taking SR's big monsters at the wrong time, etc. The goal of not having achievements in LoL at all was to prevent bad plays for the sake of metagame objectives. We need to restrict quests to trying your best to win. That means games won, the partial victories above, or wins with certain champions.