Kinda thought the point of an axe was to have a good amount of mass in the head.
A machete, on the other hand...
As for swords, granted, no one carries them around these days, but there's a reason rapiers, foils and epees came about. Against unarmored opponents, lighter and faster were a significant advantage. Easier to carry all day long, too.
you obviously have not seen

https://bladeforums.com/threads/mechaxe-titanium-alloy-axe.1640177/