I think STRless melee could work though. I am in the same boat as 'it doesn't make sense that the average person did so pathetically little damage with melee weapons' when... you know... knives and swords will kill you super dead. Assuming some other melee benefit lands on traditionally strong melee options like having str, it is a good way to solve the problem where swords and clubs are tickle fights, when in the setting they absolutely are not meant to be. Like people say "Melee being bad is realistic" but even if that were true (it is not, there is a reason the marines will ensure you exit with blades 3), it isn't bad 'because it doesn't hurt. The standard assumption is if you get stabbed once you are going to get stabbed lots, and you are going to die. If someone has you at knife point and you are equally trained and auged to them it is super weird and not good that the optimal play is to laugh at them and draw your gun.
Not to mention that it is a huge part of the SR setting that melee IS, in fact, good, and it is good enough that most of the top millitary units of the world dedicate people exclusively to melee combat and muscle powered weapons like bows.
I think there is some 'uncanny valley' in some other mechanics mind. Armor not directly aiding in resisting damage is a big 'what?' and ironically the fact you can ad-hoc it into actually doing that by spending edge to recover health is an even bigger 'huh?' But that is common in most narrative mechanics, I think the reason edge feels weird (Again, noting I have not played with the new edge and this is all gut feel) is that 6e edge is less 'narrative' than other narrative mechanics, which makes a ton of sense because it exists to replace in universe non-narrative situational mods like armor, cover, or bad weather. So you are getting these points not as narrative beats like you might in other games, but specifically for X reason, so its weird when the transfer is... kiiiinda on point but kinda not. Like I buy how my armor helps me shoot back with extra precision more than I buy my armor 'protecting' me because it heals my wounds. I think an easy fix to that though would be using edge to get the chance to auto-buy soak hits rather than heal wounds. Like, yeah that is a really specific problem, but little weird details like that can and do make games weird and almost 'gross' to play.