But Fiona being stupid is normal, there is no need to explain it.
With it being clear that the Rebellion is losing the war, with the Templars about to bust down the door and a lot of people in Thedas thinking you're possibly reponsible for killing the Divine, taking the option to let your people have safe harbour in Tevinter as free citizens, in exchange for ten years of indentured service, isn't a particularly bad deal on the face of it?
(Especially when you recall that Hawke went through a year of indentured service in DA2 to pay off the debt they owed to the people who got them into Kirkwall?)
Besides, we're told that Alexius miraculously showed up two days after the Conclave was destroyed, implying that he went back in time to before the Inquisition was even formed, thus ensuring that his would be the only offer on the table?