Not sure how the layout is on PS4, but on pS3 I can easily have all of my skills in my hotbar (including cross-class). The only thing I can't have are the potions (I'm constantly on food, however)
I've heard that before too, though. It's true that we aren't as efficient as other jobs in terms of damages, but we are not weak either. The thing is, our dps drops extremely low when we have to sing during the fight (be it for the malus of manasong / or the inactivity we suffer for casting Requiem). Of course, try to avoid unecessary moves ^^ (sounds silly tip, but I've seen that so many times before)
So when you have a mage : Requiem before the pull. That will save you some precious seconds and give the mage the most of it. Also, use battle voice as early as the mage start striking. Then I usually use my flame arrow > Raging strikes > hawk's eye > Dots > Inner release > B4B > Venimous bite > Empyreal arrow> Heavy/straight shot (abuse bloodletter whenever it's up) > Ironjaw (preferably before the Dots expire, to avoid any discontinuity). Then back to the rotation.
We have a lot more possibility as bards than any other dps. Our dps is based on procs, so keep an eye on them always. One missed proc is a drastic loss in dps. I use Blunt arrow regularly to try to prevent a cast. It has a low potency but it's a good filler anywhere in your rotation.
I notice by a quick look at your lodestone profile, you are still a fresh bard 60. Our dps at this level is nothing comparable to any other dps. At i170/180 already, you'll be able to pull great numbers. At i195+, you will love it (lots of crit gear)
If you stick to that rotation roughly, you should already be able to do quite a lot of damages.
One common mistake I see daily in dungeons : Bards spamming wide volley. That was optimal in 2.x but in 3.0 quick knock does the same thing with 20 PT less and same potency (and saving PT means more aoe)
About songs DURING a fight : They should be kept to the minimum. Try to recast Requiem when there is a little downtime (boss disapperead/nobody can attack). Peon should be used very carefully. Most melees at this level use invigorate. If it's up, they will use it after they died. A ninja can also "Goad" a player for TP. And so can an astro. Those 3 skills for TP management are instant-cast. Cast Peon only if neither of those was cast. (you don't wanna waste many seconds here again if it can be avoided).
As a healer (whm), I usually ask the bard NOT to cast manasong for me unless I drop very low (it starts to get critical around 1300/1500 mp). Before that, we don't need it. (if the healer is new and still learning mp management, they might need it a bit sooner). Obviously if a healer dies, you can start singing right away + battle voice.
Oh, and it shouldn't be worth mentionning but... wanderer's minuet is your friend. Even at level 60, some bards don't use it, and it's a shame.