You will end up at the
wiki for armor a lot.
The civilians are not wearing the leather armor because it is not part of their civvy uniform.
The "civilian uniform" will vary a bit from civ to civ but will be something like
This means that civilians will attempt to wear all of those things, of any material. A miner, hunter, or woodcutter will add their tools-of-the-trade to civvy uniform.
Note there is no "leather armor" on that list. If you used the leather to make "leather robe" and "leather trousers" then they would see it and think "Oh new stuff! Must have".
That is basically how civilian dwarf uniforms work, and you can't adjust them.
To get them to wear the leather armor as part of their military uniform, those dwarfs need to go into the military and be assigned a uniform.
Look at this other imageHere I added "leather armor", replaced "cap" with "helm", replaced "gloves" with "white gauntlets", and added "white greaves". I also added a second pair of socks and second pair of shoes and shield and weapon. If a squad, or just a position in a squad, has this uniform assigned to them, then that dwarf will attempt to wear that uniform. In DwarfTherapist, the equipped dwarf has this gear:
By giving a uniform the doubled shoe/sock/boot, you avoid the buggy thing where dwarfs will only wear one sock on their foot. The more common way is to avoid the bug is to go to each squad position and assign specific shoes/socks/boots, but that annoys me so I do this instead. You need to make more shoes and socks, but they fully cover their feet and replace/upgrade gear without my needing to tell them too.