I haven't had my coffee yet, so I trying not to sound dense on this, but I'm still not getting the argument for this. Are you saying that you'd rather be awarded edge to player every turn? That sounds horrendous to me, since players are going to get edge over most grunt NPCs in most encounters. So, if the player goes first, here's Edge for having a hire AR and better tactics. Then the NPC goes, granting Edge to the player for having better DR. Then the next NPC goes, granting more edge. In a player versus 3 Grunt scenario (i.e., the Sam versus three low-level gangers), that would easily wind up giving the Sam at least four, possibly all seven, Edge in a single round.
Is it really so wrong for the streetsam to dominate grunts NPCs in combat?
Oh no, they definitely SHOULD!
But should the player get seven edge for beating up random gangers in one round?
Well, the Sammie wouldn´t exactly get seven edge for "beating up random gangers".
It´s seven Edge for
drawing multiple attacks by the gangers while cleverly using the environment (f.i. darkness, cover) and superior gear and stats (Armor, Body, Vision Enhancements) to her advancement.
And also for taking the risk that she might not be able to easily dodge or soak all of these attacks - especially when she wants to safe up the earned Edge for a tacticool Counterattack on her own turn.
And
also, for somehow making the GM forget that Grunt rules exist

Because if the GM lets the gangers, well,
gang up , they might reduce their max. damage per turn, but will also hit more reliably without feeding the Sammie precious Edge. They might even earn Edge on their own and bring her down by attrition.