Nor would Darkspawn have to. Red Lyrium is infected too remember? It could have grown its way inside, and spread the infection on its own. It would have had around 1000 years to do it.
I posted something like this in the original thread, but it's entirely possible for the idol to have been left there by a subsequent visitor to the thaig, darkspawn or otherwise.
For instance, we know that golems were invented by Caridin in -225 Ancient, during the First Blight; therefore, they postdated the abandonment/death of the primeval thaig. But there is a golem in the thaig itself: it's in that first fight you have after leaving the expedition's encampment. That golem may have reached the thaig after the thaig was lost...or it may have been moved there by someone or something.
I think it's very unlikely that the dwarves got the date of golems' invention wrong, mostly because unlike the idol, Caridin is a figure of well-recorded history. The only reason we have to believe that the idol predates the First Blight is conjecture. I don't remember if it's Varric or Hawke who says it, but one of the expedition members guesses that the thaig's temple was built to house the idol. There's no particular reason we should assume this to be true.
Despite all this, I don't actually believe that the idol was left in the thaig by a later visitor. I simply want to point out the possibility: it means that we can't be sure that red lyrium predates the First Blight. It's certainly a tantalizing hint, but not conclusive evidence.