As someone mentioned earlier, PvP does kinda do a few things right as far as making healing exciting. Part of it is of course the random nature of incoming damage that keeps us on our toes, but let's ignore that for now because I doubt people would like wipes in PvE because a key DPS got burst down suddenly.
First and foremost, your DPS spells also provide support in a way. Whether that's helping your later healing (broil giving aetherflow and fairy gauge for instance) or helping your party with damage (glare's vulnerability up for instance), your DPS spells in PvP still manage to evoke that healer "support" flavor. This also encourages players who may wish to support the party solely to use their full kit.
Secondly, your basic attack is free. One of the issues with healing and dpsing is that they share a common resource. A common mindset I see new players get into is feeling like they can't afford to waste any mp on non-healing stuff. After all, what if they need it? Still, even at a high level I feel making at least the more basic parts of dpsing free (or even if they restored MP as one of their "aiding healing" bits) would help reinforce that they are there to be used.
Now, that all said, my ideal version of healer would have some complexity to it akin to a DPS, with support functions on the combos. Think something like how samurai or monk can keep a damage buff on themselves by returning to a given combo every so often, or Monk's greased lightning for minimizing downtime, or Ninja's huton gauge... Except instead of aiding one's personal damage it provides some sort of defensive buff to the party and/or provides resources for later healing. Heck, gunbreaker's main combo gives a shield why can't we have ours shield the enemy's target? You're still dpsing during downtime, but with a more complex version than what we have now and with the added benefit of being prepared for the damage when it does come.