Strength is also used in climbing, calculating unarmed attack rating, pushing, pulling, breaking out of grapples, preventing someone from disarming you, defending against the decrease attribute spell, and perhaps a couple more things. I have been considering awarding edge in melee situations in which one person/creature has a significant strength advantage. I like that because close combat lacks ways of gaining edge due to lower attack ratings. Any thoughts on that?
I was going to say that "Preventing from the decrease Attribute spell" is a bit redundant under the assumption that Strength
might be a dumbstat, since it only protects you from decreasing itself. Then I realized that you still collapse when a mundane Attribute reaches zero (Wow, I just noticed that
Decrease Attribute is quite a tool against dumbstatters in general...

). You can also add Strength requirements for Machine Guns to that list.
So yeah, the problem isn´t really that Strength is a "Dumptstat" (it isn´t), it´s the fact that Strength scales with unarmed combat for both AR and Damage Calculation while having no impact on melee weapons at all. A Strength Score of 5 is already enough to compete with most melee weapons, often making them obsolete alltogether. And that´s without bringing stuff like Dermal Deposits and Bone Augmentations into the mix. And bear in mind that melee weapons have first to be readied (and hidden and public) while fists are always available.
I´m not sure if it´s going to cut it, but your suggestion is surely better than leaving Strength completely out of the mix for armed combat. If you want to go the "edgy" route: Why not do the most obvious thing and add Strength to the baseline AR of the Weapon? Most Melee Weapons have Attack ratings that are so low that one can only wonder if these were originally supposed to have Strength added to it and then someone in editing convieniently deleted a sentence or two...