If there were no Hawke, Flemeth would have found another chump. After all, she had time after she sent Morrigan away to go find someone to take the amulet to Meratheri in the Marches.
As to Corypheus, I doubt that factored into Flemeth's thoughts at all.
Also, the chances of Corypheus being the Elder One are in the single digits, in my opinion. It's too easy an explaination. Besides, the Elder One shouldn't be some confused tainted magister. He should be a new entity entirely. Corypheus seems to be the Saruman to the Elder One's Souron.
I agree with your first sentiment. Whoever is charged with delivering messages and packages in Thedas is pretty damn good at their job considering a letter can get from Lothering to Gamlen's hovel quite reliably (Bethany Hawke's correspondence). Flemeth could have simply hired one of those guys.
Corypheus is a sideshow compared to the Elder One, but his role is important. Think about it, the Inquisition strives to be a massive paramilitary organization spanning the whole of Thedas. That pretty much describes the Grey Wardens to a tee. Despite their obligation to counter the Darkspawn there is nothing stopping the First Warden from turning the full might of the order to fighting this new demonic threat, which is certainly as dangerous as the Darkspawn. With Corypheus having apparently caused some unrest or even a full-blown civil war among the Grey Wardens, it really does fall solely on the Inquisition to fill the void left by the templars and mages in the wake of their own war.
As for Hawke, the Champion really was just a bystander in all of this nonsense. There was not a signal substantial thing we could have done in Dragon Age II to change anything. Even if we had killed Anders for no given reason early on things might have still popped. Why? Well, there are more mages now than before. It has been bandied about in some tavern talk back in DA2 that there has been an unusual population boon of mages in recent years. This means there are more and more mages that need to be penned up in Circle Towers. I imagine that when there was only 1 mage for every 10 templars, they could easily be controlled. But what about 3 mages for every 10 templars, perhaps more? Then things get interesting.
The Chantry was playing with fire from the start, they were priests trying to control an army of knights with addictive lyrium treatment who were in turn charged with controlling mages who range from apprentices to seasoned Enchanters who rival Tevinter Archons in power. Now the whole house is burning....