I think you're greatly under appreciating the songs. Bards are by far the best utility dps in the game and songs are a big part of it.

Mage's Ballad can "turn the tide of battle", because when your progression fights drag on that ballad can mean the difference between a wipe and a kill.

Foe Requiem is even better, it dictates where the tide turns. The damage buff is extremely powerful, and you don't lose 20% damage playing it. That's free damage.
Mage's Ballad can be used to save a party, but Foe Requiem is a constant support otherwise. You should be able to keep doing it and stop to save mana if you know you'll have to help the healer's mana.

You said you only need Mage's Ballad for Titan's heart, right? But then you say you don't play Requiem much cause people always ask for the Ballad.
You should be using Requiem as much as you can. It even buffs your own damage, but it's bigger deal if you have a mage.

Army's Paeon is not as useful, I'll admit. However, no one should ever auto-attack for 20-30s to do normal dps for a few more minutes. That's bad and will probably kill your group in the long run if you're doing coil (or even Titan's phase 5, maybe).
It's not as useful, because it's usually not needed since all TP users have Invigorate. However, when it is needed bard's are the only ones that can and should provide. There are times when this can also turn the tide of battle when you're pushing an enrage in T4 or even just trying to get the kill before you all die.

Of course mistakes like that go away as people know the fights by heart. But that just means bard songs shine in progression fights.
They're a safety net that can help you push through odds that you aren't necessarily ready for otherwise.
If that's not turning the tide of battle, I'm not sure what is.

I don't mind bard's having all the utility it does, while pulling decent dps and being extremely mobile. But you really can't ask them to be any more important.
They're the only required dps in a group already.