League-of-legends – How long does a stack of the Spell Weaving mastery last

You have one false assumption in your question: what you observed was not two quests, but rather only one.

In other words, the quest was "Capture Point X, while defending point Y from Capture". The quest lasts until one team manages to capture the other, which means, in your case, the 'defend' portion of the quest lasts until your team manages to capture the enemy point in question, rather than a simpler, "defend for X".

You could theoretically have to defend that point for the rest of the game, as long as neither of the two target points changes hands.

Smite has a cooldown of 70 seconds. Baron takes 7 minutes to respawn, the dragon takes 6 minutes. This gives you 6 smites within a 7 minute time span, or 4 extra smites you can use whenever for a free 10 gold. Knock out one of the siege minions in a lane, kill a random Golem... all cases where you'll also get some free gold. As the cost is an ability you can't even use in team fights in most cases (barring Annie and others), it's essentially around 120 free gold or more in the period between objectives. Even a farmer who would already get the kill, it's still an extra 10 gold for virtually no significant cost - it's fast, you barely need to travel, no mana cost.

It's not a lot, ultimately - it translates roughly to the equivalent of 1.4 gold per 10 seconds if you were to fire it off every time after cooldown (and ignoring the periods in taking down the objectives). But it is pretty effective and can give you a pretty penny, so if you have the spare mastery and are going with Smite to begin with, it can be very functional. Just realize that you can toss it out around 4 times between objectives, give or take based on the obvious desync for Dragon/Baron spawns.