It's all a matter of checks and balances.

Should BLM deal more damage than a... let's say MNK?

Yes, it should. That alone is not a problem in itself, because the BLM should be limited by mana. So the DRG (or any melee) should be able to DD indefinitely, while the BLM should, at some point, wait for MP or have some support class help him regenerate it.
The problem is that right now the support class (BRD) is also a ranged DD, which can deal damage close to a melee DD, with the added benefit of increased mobility.

Simply not great design. In fact, I'm strongly of the opinion that the lack of support classes/roles hinders greatly the depth of the game. Hopefully that will change, even though I won't hold by breath for it.

IMHO a way better setup for the game would have been:

Melee DD
Strong points:
- sustainable damage over long periods, less limited by TP --> they can favor DMG+ gear

Weak points:
- more in danger's way
- more stressing on the healers

Ranged physical DD
Strong points:
- increased mobility
- less stressing on the healers

Weak points:
- limited by TP, with their abilities costing more --> they need to favor some TP+ gear, depending on the party composition (support or not)

Ranged magical DD
Strong points:
- very high spike damage
- AOE damage
- somewhat increased mobility (have to stop to cast)

Weak points:
- limited by MP --> dependent on support roles and/or gear for very high damage in the long run
- low HP/defense

Support classes (like BRD should be)
Strong points:
- powerful buffs/debuffs
- increased mobility

Weak points:
- limited potential for damage
- in between HP/defense

However, we ended up with a class that has Ranger level damage and does also Mage's Ballad. Of course 5 BLM + 1 BRD will always be better than 6 x MNK for pretty much anything, unless a single target LB is needed.

-Lak