So, my two cents as someone new to the forums (but not the game), if I were to re-write the unarmed rules, I'd do something like this:
1. Give Unarmed Attacks a baseline damage rating.
Probably 2S, putting it on par with a Sap (which you can still use to take advantage of a Clubs specialty), a Knife (which makes the damage Physical and lets you use Blades and can still be thrown) and the Bike Chain (which still gives you a free Trip action). So these weapons aren't invalidated and have uses, but unarmed is competitive. Let Dermal Deposits still give a +1 to unarmed damage (free Knucks!) and Bone Lacing/Bone Density still gives its listed bonuses. A troll with Titanium Bone Lacing would be hitting as hard as a Combat Axe... and I'm okay with this. Similarly, putting Hardliner Gloves on a troll isn't going to make him hit any harder, just let him hit more lethally... and they won't be of any use to someone with Bone Lacing... but I'm okay with that too.
2. Make Strength the Default Close Combat Attribute.
This means that while the damage is baseline for unarmed combat, it will let your Strength be the determining factor in how hard you actually hit someone and damage them since it flows into the Base Damage + net hits system that every other combat method uses. So stronger characters will have a better chance of getting net hits. Special actions like Wrest and initiating a Grapple can still use Agility + Close Combat because, honestly, those actions still make sense as being about coordination and agility since you're not trying to hurt someone, just get a grip on them (and no matter how good a grip you get, prying a weapon out of the hands of the high Strength troll or actually restraining the high Strength ork is going to be pretty hard since those are contested by Strength).
3. Do away with Close Range Attack Values for most weapons.
Melee weapons with a Near Attack Value (Knives and Combat Knives for instance) would still have their listed vlaue. And some exotic weapons like Bullwhips and Monofilament Whips can (and should) break this rule with their own attack ratings due to the nature of their use. But what do you replace those Attack Values with?
4. Have the Strength + Reaction calculation for Attack Value apply to weapons classified as Unarmed.
So giving Unarmed Combat a baseline damage solves the problem that putting on a pair of Hardliner Gloves makes a high-Strength character do less damage. This change makes it so that those same Hardliners don't make you somehow less threatening. Knucks, Shock Gloves, Bike Chains and cyberlimbs with the Hardening or Shock Limb accessories are using the same Strength + Reaction calculation to maintain the same level of threat, with the weapon you're using determining the base damage.
5. Use Agility + Reaction for Close Range Attack Value of other melee weapons.
This is where the elf with a katana is going to excel. You don't want to neglect Strength all together (anymore than a character who plans to use guns should completely neglect Agility), but the Agility based character is going to have an advantage in generating Edge to reroll dice, bump up their rolled numbers, use Edge Actions like Shank and Knockout Blow and add their Edge to their attack pool to compensate for a lower Strength for some potentially devastating hits. A Strength based character can compensate with an average Agility (something the pure Unarmed character wouldn't necessarily need) and a strong Reaction (something a good combat character of any sort should probably have for Initiative purposes anyway). But the Agility character is going to be more likely to gain an advantage and parlay that into results when using a weapon.
6. Exotic Weapons can break these rules.
Monofilament Whip is a great example of that already. It should absolutely have an Attack Value of its own due to its nature. And it's a weapon that should probably use Agility + Exotic Weapons to attack rather than Strength, which can easily just be specified in the description of the weapon. Bullwhip is similar and should probably be Agility + Exotic Weapons and keep its default Attack Rating and let you get free Trips. Exotic Weapons would just need to have these spelled out in their write-ups... but they're Exotic and special already, right?
EDIT:
Slight addendum I thought of after writing this up yesterday. I realized that replacing the listed Attack Ratings of melee weapons with a calculated Attack Rating like unarmed has makes a Combat Knife and a Sword effectively the same (except the combat knife can be thrown). I'd also dial the Combat Knife to 2P and the standard Knife down to 1P. The 1P Knife is still useful because using it takes you from a 2S unarmed attack to a 1P armed attack so you're doing actual Physical damage, plus you can use a Blades specialty and throw the weapon in a pinch, while the Combat Knife is getting everything I said about the regular Knife above.
All the other weapons currently listed work just fine as is because even when the damage is the same, there's value to the weapon as it stands. Katanas and Polearms have the same damage potential, but there's a big difference between wielding a katana and a polearm. And when you get to Clubs, a Staff and Telescoping Staff have the same damage value, but... they're the same weapon, so okay. (As an aside, I do question the COST of a Blackjack, but even in the RAW, I question 75 nuyen for a bag of sand.)