Why don't Paladins have any unique "Cures"

[Enkidoh]

You mean back when people expected us to do part of the healer's job? No thanks.

PS: PLDs are not holy knights in this game. They're the bodyguards of the sultanate.

^This pretty much. The reason PLD lost it's healing abilities in ARR is probably due to the fact that SE added SCH as a healer (which didn't exist in 1.0), thus freeing PLD of that responsibility and allowing them to concentrate on tanking. In 1.0 there was just CNJ/WHM in the healer role, so PLD was made into a hybrid pseudo-healer/tank simply out of desperation.

And as Duelle said PLD are not holy knights, but members of the Sultansworn Elite, the royal guards of Ul'dah. Also considering holy as an element really doesn't even exist in FFXIV (okay, there is the 'astral' alignment, but that's not really an actual element in the true sense), it's kind of moot to even call them 'holy knights' (as someone else mentioned, PLD is actually simply called 'Knight' in the Japanese client).

Having said that, I'm certain SE is keeping missing abilities and spells from version 1.0 in reserve for future level cap increases, so who knows what might happen once Heavensward is released and level 60 is reached (and levels beyond) - PLD just may get back it's healing abilities after all, anything is possible really.