I'm not going to say CNJ should be the sub-class or what it would take to make that work.
Look at it this way, though. Bard is really exceptional in what they get from cross class and I think comparing what it would be from what it is now is a mistake.
All the other dps jobs get very little compared to the bard. I think mostly utility is by design since bard is such an exception.
We can agree cooldowns are a big deal, right? Bards get TWO. No one else gets two from cross class. On top of that, bard has its own.
NOTE:
I know bard dps is bad and they already need these to stay competitive but they've already said they're going to nerf them and that they get too much cross class benefit. I am just talking about the overall design rather than numbers (which they should adjust with any change).
So for example, monk gets a cd from lancer, like bard, and invigorate. Lancer gets internal release and mantra, like bard. Both use marauder otherwise and get what? Mercy Stroke and fracture. Only monks use the latter (and only when tp is fine) and the former is underwhelming.
SMN gets a cd from bard and none from THM, it also gets swiftcast. THM gets a cd from bard and has none for itself, but it does get virus and quelling strikes as well.
Bard has its own cd and takes from MNK and DRG. It also gets Invigorate (which is amazing), mantra (mini-mantra) and Second Wind.
It's really the only dps that really benefits from having two other dps classes as cross class. The only other is SMN which gets only swiftcast and surecast from BLM anyway.
IS THAT FAIR?
Well, yes it seems fair. Bard damage is the lowest and cooldown management is like the only complication for the class. It's balanced well like that.
So what's my point if I think so? Well, I'm just saying it's not hard to imagine a world where bard is like every other class in gaining mostly utility from their cross class and one major damage increase.
I, personally, don't think they need to nerf bards. But they are, and that's just the case. Numbers can be balanced if they change the job requirements OR if they simply take IR or BfB from bard.
Having one class for a damage buff and one for utility is pretty par for the course.
Damage decrease on movement would be a much better solution though.