For what reasons would you train with a sword over a mace in the late medieval and early Renaissance world?

by He_Coontbuster

According to images I've seen and commentary I've heard, it would seem that swords end up being used in a way they were never designed to be ( like with half swording and the murder stroke ) or fights with swords would devolve into wrestling and bashing heads in - yet they still evolved and remained in use.

Why would someone train with and use a sword when they were nearly useless against the armor of the time? Why didn't everyone switch over to blunt weapons?

Edit: some grammar

WARitter

I deal with the overall context of swords in the later Middle ages here. In short, swords were easy to carry and useful in a wide variety of situations. Please feel free to ask any more particular follow-ups!