Would it be viable to play pandemic with an "eradicate all diseases" win condition?
You can get a pretty good idea about the extra time needed just by looking at the board when you finish a normal game. How many cities still have cubes in them? How many cards are left in the player deck?

As an absolute lower bound, you'll need two actions per city (one to move, one to treat), or one for the medic. If you've got three players and one is the medic, that's 8 cities cleared per six cards drawn.

But usually you'll have more than 8 cities, you'll need a lot more than one action per city to get to all of them, and the infection phase will be undoing half your work. You'll also usually have 10 or fewer cards left in the player deck. The odds of being able to clear the board are really, really low - and worse, if you can do it, it's probably just because you're lucky about infected cities being close together.

As others have said, simply adding an expansion and varying the difficulty using the existing means is probably a better bet.