Bard's DPS scales immensely off of critical hit rate, which SCH, AST and DRG can all provide to it. Combined they could provide up to 35-45% (if AST caters to BRD specifically) crit rate at the moments BRD wants it the most. To put it in perspective, you'd need the relic weapon to get more than that amount of crit rate on top of running for BRD's Best in Slot gear everywhere else. Just having a SCH and DRG in your party was enough to more than double your crit rate at the beginning of the expansion too.
Like MCH they also benefit from DRG's near 100% uptime piercing debuff, but boosting their spike window is far more important to pushing their overall DPS higher. Removing piercing wouldn't undo the requirement to have a DRG to maximize the potential of a BRD.
A better way to balance it would be to shift the procs to either flat rates, or direct hit, provided the latter was never given another raid buff on another class. Given SE would likely break that rule via an oversight in the future I'd prefer the flat proc system, and add a buff to proc rate in Raging Strikes, and maybe a trait that boosts every other Raging Strikes a bit further. Basically, give them more personal DPS to make up for the nerfs.
I don't think it's unreasonable to nerf some of BRD's raid dps either. Foes could easily be reworked into another song for their rotation and give them a different option. They kinda need an AoE song other than Ballad, and they could use a buff to Paeon too. I wouldn't mind a song that boosted their GCD damage too. Battle Voice could survive a 5% Direct Hit rate nerf while still being good.
I think nerfing the bardsong buff itself is probably what needs to happen though. 2% crit rate with 100% uptime has a weaker but similar effect as Piercing, but for your whole party, making it immensely, if deceptively good. Moving that to Direct Hit would help bring it down, limiting its overall power. You have to keep in mind that while the crit raid buffs only affect rate, their power still scales with item level far better than Direct Hit or % increases that aren't as high as TA, since boosting the Crit stat with gear boosts your rate AND the damage that your critical strikes do. Direct Hit rate buffs always give the same DPS boost, regardless of what your stats are.
I don't feel BRD needs GCD songs at all either. I'd rather remove them than buff them. It would be healthier for the game overall to not push BRD's utility further too (They're running out of nifty things to add that don't explicitly beat MCH further into the ground). Instead give it personal cooldown boosts that buff its spike DPS to match what it was capable of with 2 crit rate buffs while retuning its core kit to not function on crit at all anymore, undoing what makes it broken. MCH should get all of the love it frankly needs and deserves on top of pushing ahead of what BRD is currently capable of.