I've found two situations where healers are fun to play.

1) Everything is going wrong in casual content and healing becomes a fast-paced resource management game. Shinryu normal was great for this before everyone overgeared it because the average DF player would die frequently. I fondly recall a 31 death clear that included two healer LB3 and snap decisions between who gets healed and who dies. This sort of engagement stems from a rare combination of underperforming players, a lack of hard wipe mechanics, and a skilled tank to keep the boss off the healer.

2) Early prog in 'hardcore' content. This is all a learning and resource allocation game. When is the next raidwide? How hard does it hit? Do we need shields or can it be mitigated safely? Which oGCD do I use there? Should I use a GCD heal and save that oGCD for a movement heavy mechanic in 20 secs?

The common theme is having to adapt to unknown situations. Clean clears are routine, and healers' kits are so basic and predictable that routine rapidly becomes boring. It's unfortunate that the majority of content in this game is all identical for healers. Damage from trash in dungeons is very standardized. Tankbusters and raidwide frequency is similarly standardized in dungeons and normal content.

I will mention that SE has been changing the shape of damage in recent patches and that it has made a big difference in how progging fights goes. The opening mechanics of WoL normal and E12S P2 make a nice change from opening with a heavy raidwide to strip away preshields. Attaching a heavy bleed to the raidwide in E11S changes how a healer approaches raidwides in that fight because the bleed will often expire as the subsequent mechanic occurs. Neither of these are novel mechanics, but it's still good to see that SE is at least trying to change how healers play. It's just a shame that they are refusing to budge on the >90% downtime issue that condemns healers to repetitive, dull gameplay for the majority of the time.