Pretty much. Your best bet is to set the uniform to replace clothing, and only use exact matches. That way whoever is assigned the uniform is forced to equip the right armor.
The dwarves should strip naked, rush the armor stockpile or forge, and equip what they need
This.
The reason they are only equipping the chainmail is, most likely, that the rest of their outfit is already "full."
This page has a tl;dr breakdown of how the game determines what and how much armor/clothing a dwarf can equip, and why. Read up and understand; that will solve most equipment issues you'll ever run into.
I'll save you a little time and address the two main issues here: size/layering and [SHAPED] pieces.
1. Any given body part can only equip a certain amount of apparel, and that amount is also affected by what specific types of clothing are chosen. Just like real life, a DF character can only equip so many shirts, and fewer coats. The dwarven standard outfit (in a vanilla game) includes plenty of space for armor, so that is likely not your problem, but possible.
This thread includes some exposition about that specific issue.
2. Any given body part can only equip one [SHAPED] item at a time. High and Low Boots, Greaves and Leggings, Breastplates and Leather Armor, Gauntlets, Helms and Caps are all shaped,
including cloth or leather items. This means that if your soldiers are wearing cloth caps, they won't equip a helm unless you set the uniform to "Replace Clothing."
Setting them to Replace Armor will force them to equip only the items they have been assigned through their uniform. They won't include anything else, so make sure you have at least footwear, legwear and something for the torso or they'll get bad thoughts for lacking those clothing items. Or just assign them cloth shoes or whatever until you finish those parts of the uniform.