It should also be mentioned that none of the BRD's songs are universally useful, like a straight +det, +dam, or +crit buff would be. BRD either helps out healers (with Ballad, since BLM and SMN are both effectively infinite mp), caster DPS (with Requiem; BLM damage is pretty much all elemental and all SMN pet damage is elemental even if all of the SMN's damage is unaspected; it buffs BRD damage a bit, thanks to Windbite and Flame Arrow, but that's not a huge amount), or DoW classes (with Paeon).
The only other class that could be labelled as "support" is SCH, when they elect to use Selene instead of Eos. Selene can buff *either* skill speed or spell speed at any given time (if you're good, you can make sure that one of those two is up at all times) so it's not universally useful either. On top of this, just like with BRDs, using Selene decreases your healing output (Eos packs the +healing buff and Whispering Dawn) so it comes at a cost to your primary effectiveness when you opt to use it.
Using the existing examples of "support" that we've got, there are 3 criteria that it needs to follow: limited uptime, benefit to a subset of total classes rather than all, and it must come at the cost of your primary functionality when it's active. There's kind of a fourth pseudo criteria which is that support isn't a primary role so you must still be a fully functional tank, dps, or healer.
If you really wanted to play up the random aspect as part of Corsair (not entire sure I think that the piratical, gun-using job should lead into what has classically been "Gambler" but whatever), you could have each of the effects provide a static buff to a random subset of classes. The "cost" could come as the dice applying a debuff to you that reduces damage by 20% for 10 seconds after it's used. If you wanted to go the route of multiple types of dice, you could have...
Restful Dice: throws 2 dice that have half of their faces as the DoW symbol and half as the DoM symbol; restores 100 TP and 10% MP to everyone if one of each face is shown, 250 TP if it's double DoW, and 25% MP if it's double DoM
Debillitating Dice: throws 2 dice that have one of the elements symbolized on each of their faces; reduces target's resistance to the damage types shown by 10% and reduces resistance by 25% if they both show the same damage type
Corsair's Dice: throws 2 dice that have each of a sword, a shield, a wand, a starburst, a tree, or a lucky 7; sword increases attack power by 15%, shield increases defense and mdef by 10%, wand increases attack magic potency by 10%, tree increases healing magic potency by 10%, starburst provides a small but visible increase to the limit gauge; if 2 faces show, they increase the benefit by 150%; lucky 7 only works when you get 2 of them but it provides the benefit of all faces as if 2 of that face had been shown
Extra Die (Lv. 50): A long CD ability that causes your next Dice ability to throw an extra die (3 dice of the same face provide no greater benefit than if there were 2 dice showing that face)
The 5th job ability would likely need to be an actual attack.
The BRD construct (which this follows: 3 support, 1 long CD ability that improves those support abilities, and 1 attack) is a pretty good one. You'd need to give the abilities pretty substantial costs (like consuming 75% of your max mp) and/or long CDs, but the randomness of the benefits means that you could get away with them providing more than the other support jobs are able to provide.