DPS class with the fewest buttons?

[Allyrion]

NIN - slightly easier than the other two melee but the mudras waste save space on the hotbars

Fixed that for you.
Mudras are an anti-button bloat mechanic (it's flaws are with lag not bloat).
Imagine if the ninjutsu skills were each given individually. That would waste space.

As for the geniuses saying MNK has no oGCDs:
IR, B4B, Invig, Mercy, Mantra, Shoulder Tackle, Elixer Field, Steel Peak, Howling Fist, Perfect Balance, Purification, Tornado Kick, Meditation/Forbidden Chakra (13)
DRG:
IR, B4B, Invig, Mercy, Mantra, Jump, Spineshatter, Dragonfire, PSurge, LSurge, Battle Litany, BotD, Leg Sweep, Elusive Jump (14)

They're pretty much the same. If you want to argue that Tornado Kick (for example) isn't used much we can do the same for Elusive Jump and DRG Mantra. Neither of them are easy on buttons considering you're still WASDing the hell out of your keyboard at the same time.

Not just Tornado Kick, but Purification and Forbidden Chakra too.
They're not guaranteed oGCD skills except for the first Forbidden Chakra.

You might as well not separate Purification and Forbidden Chakra separately because Meditation stacking is the main lock.
It's disingenuous to separate them. If you used a Purification then you didn't use a Forbidden Chakra.

Perfect Balance is also not exactly like the rotational buffs either.
You usually use it to get back GL3.
Doesn't really matter if it's on or off the GCD at that point (except for timing), because you don't weave it between your regular GCD rotation.

MNK uses Shoulder Tackle, Elixer Field, Steel Peak and Howling Fist (4) regularly.
DRG uses Jump, Spineshatter, Dragonfire, PSurge, LSurge, Leg Sweep and BOTD (7) regularly.
Not even counting Battle Litany.

NIN uses Jugulate, Mug, Ninjutsu (shared cd so I'll count as one), Dream within a Dream, Kassatsu, Duality and Trick Attack (7).
That's not even counting Goad or Assassinate (since they're niche).

I'm obviously leaving out the shared damage cooldowns that are equal anyway.

But just counting abilities is a flawed comparison.
NIN is obviously the most oGCD active because of ninjutsu every 20s in addition to the rest.
A proper comparison would require you to take all the recasts into account.

DRG is the second most oGCD active.

While MNK has less to deal with, it's still a lot more now that Shoulder Tack doesn't have a ranged req and Elixir Drive is only 30s recast.
But Forbidden Chakra (and Purification) and Tornado Kick are really fight dependent.
I like how MNK plays, but those skills just don't fit with the regular oGCD wielding of the other melee.

Which is fine, because MNK isn't designed around oGCDs as much as the other classes are.
It's designed around GL stacks, positionals and (now) filling downtime with meditation.