Well I originally decided I'd actually grab a few easy weapons from the trolls in the cave north of burthorpe.. but after nearly dying to one I had to go get food.. came back, barely got a kill and no drop. Repeat, still no drop. I can barely kill a chicken without losing half my health.
So my first noticeable level was.. 10 cooking
I've killed a few things now.. and I'm starting to notice the fact that I'm getting literally 0 drops other than raw rabbit..

After killing a bunch of chickens and a few more rabbits.. I vaguely remember something about not being able to get drops while unarmed. I see nothing else stating that on the wiki or anywhere on google.. but nothing is dropping. Even "100%" drops such as bones. The only thing dropping is raw rabbit, but the rabbits have a max HP of 1 and I seemingly can't miss, so I think they're treated differently.
I got 4 attack, and by now I'm pretty certain you can't get drops unarmed from the majority of enemies. My first thought of how to solve this was to just do rune mysteries, craft some runes, and maybe if I cast then things will drop items again and I can actually obtain weapons. The issue with this, is that I'd have to use the air talisman I got from the quest reward.. which breaks my rules.
I then remembered that the enemies in the Blood Pact quest drop weapons. Maybe because they're quest enemies they'll be similar to rabbits. So I headed on over, anddd.. Xenia won't let you into the room unless you equip the bronze dagger she gives you. So I simply equip it, throw it away, and enter the room. I didn't technically "use" the dagger, so I think this is fair game. Hell, you spawn with a bronze dagger equipped, so if that's all it takes then my rules are physically impossible.
So onto...

The fight. 75 health.. is a lot, when your max hit is 1 and you rarely hit. Luckily he also only hits for 1, but with 100% accuracy. So after 19 cooked rabbits and 3 crayfish, eventually..

I finally have my first weapon! Kayle's chargebow. I finished the dungeon / quest and also obtained Caitlin's staff, Reese's sword, Reese's off-hand sword and a Combat XP lamp. I'm going to be destroying all of them except for the main hand sword, including the lamp. Technically the weapons were all drops.. and I don't think there was any other way around this issue.. but I'm still going to get rid of them all anyways.
I decided to check for my own sanity.. and yep. Cows now drop bones, cow hide and raw beef. Chickens now drop raw chicken, feathers and bones.
I ran over to the trolls in attempts to get a quick weapon, and.. didn't get one for a long time. I got a bunch of drops that weren't even on the wiki, so I'm not even sure if these trolls can drop weapons.. decided to go kill goblins instead and got this on my first drop:

So now I can safely destroy my sword. I also got.. an air talisman, a staff of air, bronze bolts.. plus I got bronze arrows from trolls earlier. So now all of those are unlocked, not sure how much use they'll get though considering this chargebow doesn't need arrows.
Somehow I forgot to include in the OP that I was going to allow salvage to unlock items - for example, I just found a tiny plated bronze salvage, and that would have allowed me to roll an RNG generator to decide whether I unlock: Bronze armoured boots, bronze gauntlets or a bronze med helm.
I might still allow this method.. but for now I'm really not sure it's necessary, so I'll attempt to get by without doing so.