Yes,we may never know for sure,but it's not impossible that this tool was shaped deliberately,possibly for a single specific job.
It wasn't not uncommon in the past to custom re-shape a tool like that.Like when people carving a dug-out boat cupped an axe blade to make a side-action adze of sorts(often making those in pairs,left/right).
As to the metallurgy(my quote function just went off):
Keeping the edge cool while heating is not hard,the trouble is that you can't very well manipulate the forging with one end stuck in a can of water...And as you take it out to do stuff to it,the heat will rapidly soak towards there...(steel is a good conductor...).
All you say makes good sense,right on.
Just remember that heating will create an added difficulty by making the heated parts deform much readier than cold,so if heat is not distributed Just right,things may begin to go out of whack in a not necessarily controlled manner...
Conversely,a cold forging will distort evenly,that only being differentiated by sectional dimension.So,one less factor to juggle.