They started the Inquisition to end the mage rebellions, restore order, and re-establish the Circle of Magi. The wanted the HoF because they united the races of Thedas and ended a Blight in only 1 year(the shortest blight ever) with no Warden aid aside from a junior member, and Riordan. They are the most qualified person in all of Thedas to serve as Inquisitor.
This, exactly [and before "HoF would have died at Cory's hand"--yes, we know that now, but Cass and Lel didn't know that at the time. In hindsight, it's good the Warden wasn't around].
Proven track record with dealing with mages and abominations at Kinloch Hold (saving Cullen's life in the process, as a side note), and also killing the Archdemon (who was not "just a dragon", to answer up the thread a bit, but also the vessel of an Old God)--and, if they didn't sacrifice, being the only Warden in five Blights to have survived doing so. On top of that, they picked the dwarf king and settled the matter of the Ferelden throne, ending a Civil War (and have a decent chance of also being a political animal, if a Cousland--and/or Queen or king-consort themselves). They managed to get the Dalish to join in (or, alternatively, bring on freaking werewolves!). They may have also managed to get the Legion of the Dead to the surface for the first time in a very long time. On top of that, they are the Commander of the Grey in Ferelden, as well as the Arl/essa of Amaranthine. So, proven political track record. If the HoF romanced Zev and stayed a Warden, s/he also has connections to the Crows (and may, in fact, have acted with Zev as a leader in the Crows for a short time). And, of course, s/he was assisted by the notorious Witch of the Wilds (which Cass did know--even if Lel wasn't a part of DA:O, cf:
Cass: You expect me to believe Flemeth helped him/her [Hawke]?
Varric: Come on Seeker, do I need to recite the Tale of the Warden?
They found the Ashes, and the Temple. Ended an Undead threat in Redcliffe against all odds. Saved Eamon--and may have saved Connor and Isolde as well. And may have dealt with a bunch of undead at Warden's Keep.
They know how to deal with mages and the demonic forces that accompany them. And Alistair is also an ex-almost-Templar.
In contrast, Hawke made a bit of money, bought a house, ended a war with the Arishok and was partly responsible for the chaos in Kirkwall. And freed Corypheus. And irritated Orlais by killing Prosper de Montfort. And needed Cullen to step in to help save them (and thus the Circle of Life is closed, with the HoF ultimately being the reason Hawke's life was saved there
)
The real question is, why did they bother asking for Hawke? 