The thing is that Edward is the only first tier unit who can double constantly stuff and can claim free kills with the brave sword.
Even if he got bad levels in the first chapters, he still can catch up.
1-4 and 1-5 (wind edge on the ledge) are good chapters to give him some levels.
Also it's not that other units are less fragile than him.
Nolan is only marginally tankier than Edward. He'll have problems to take two steel axes, much less two tigers.
Aran has really bad start and his bad speed growth is another problem for him.
If you play without transfers a semi normal leveled Edward beats Jill in HM easily.