There needs to be context to this. Should I be singing foe rekt at every moment of the fight? It depends on the circumstances; where there are downtimes and you're better off singing ballad to give healers a bit of mana, and to save up mana for foe rekt for phases where having a higher dps could be more helpful (to minimize the time spent in that phase)
If a healer is running out of mana doing something, then more often not then something has gone already wrong (requiring raises, overhealing, or just poor mana management). Not to say where there are moments when none of that happens and they do need the mana, but it's so far and few inbetween.
I think someone mentioned this before, that is on the fault of the MNK/DRG dying if they needed a raise on the first place (and thus TP). Now you have to weigh this. What's going to be more dps, a bard having no damage penalty using their AoE, or a bard sarificing 20% of their damage to increase TP regeneration for someone who just got raised with a 15% stat penalty? The healer is probably low on mana after raising that MNK/DRG; should I use ballad over paeon? If it was an Aoe intense fight like T4 and such, then yes I would use Paeon, but that is not the example you provided.
A lot of the bullets you made seem to fault more to the other party members which funnel it to the bard, such as the healers needing mana (with very few exceptions), or the MNK/DRG needing TP after a raise. I mean all of the songs are pretty circumstantial with the exception of foe rekt, but even that needs some context to when you should play it during a lengthy boss fight to manage your mana and increase dps when its really needed.
I have yet to have an instance where people in my party dies because I wasn't singing. It could have been a contributor (such as the healer running out of mana), but that also stems from other problems (like them having to spend mana on raise in the first place).
This is about it really. If you're not going above and beyond your role, you're nothing special (good or bad). If you try to do that but fail at it (letting the tank die in the healer's case, or playing the wrong song at the run time), the party suffers from it and you're probably better off just sticking to the ground.