ValhallaGH
Explorer
According to my experience players begin to start their own side "plots" if they are not occupied with the game.
Really? Then I wish you'd have some experiences like I've had.
I've had players create, on their own initiative and out of their own resources:
academy to teach villagers how to resist rape attempts (known as the "Anti-Rape Army").
purchase lots of slum areas to create cheep, quality housing for refugees (of Cyre).
start charitable organizations in the midst of slums, and defend them by wiping out several local gangs.
a major political faction, based around their character's seduction of a noble's daughter.
shipping companies to trade across oceans, employ some NPCs they defeated, and generate wealth (for their mansion).
pirate companies to employ some defeated NPCs and generate wealth.
... and others that I'm blanking on.
Each and every time it has added a new dimension to my campaign, made things even more fun, and been a great story that any of us can recall with a single word or phrase. They were enjoying the campaign already, were inspired by what it contained, and decided to do something interesting and fun that fit both their character's goals and their own personalities. The results have been awesomely fun, and just made my games that much better for everyone (especially me, since the whole thing was a surprise that I got to learn about as it unfolded, unlike the main plot).
I've had a few bad ones, mostly a single player wanting to make his PC's personal story into the campaign's main story (instead of simply a developing part of that tale). The secret prince of the destroyed nation, the gunslinger with a grudge against a particular rail baron, and a couple of other character concepts with a built-in story that the player wanted to make into a major campaign focus.
None of those were started in play. They all happened in character creation.
So, to the OP, go with it, see what happens, and then decide if you need to do something about it. You'll probably enjoy it as much, or even more, than your players do.
Good luck.