This is just my speculation, but it's probable the reason why SE did FATEs this way is because in FFXI when Campaign was first released, players discovered a simple exploit in healing during Campaign battles, in that all they had to do was simply sub BRD and then sing songs over and over again until the battle ended, without actually doing any actual healing of other players or fighting in the battle, and got lots of EXP and Allied Notes for it.

SE later quietly nerfed rewards for any kind of defensive participation in Campaign to put an end to this exploit in an update, in that healing spells indeed no longer gained any real EXP in Campaign, so it's highly likely they've carried this way of thinking into the design of FATEs (don't forget a lot of FFXIV's development team came from FFXI). So yeah, as usual, the old meme 'this is why we can't have nice things' springs to mind here - blame lazy and stupid players for spoiling it for everyone else.