I'm sorry, but this feedback is not meant to open discussion based on the balance between healers, it is meant to open discussion about how the game systems of The Feast drive people away from playing the healer role, and my suggested fixes to make healing (and therefore learning how to heal) a bit less insane. So naturally, I thought you were speaking in terms of how white mage is too hard to deal with from the perspectives from all jobs, not just the other healers.
That said, yes, White Mage is the strongest of the three healers dude to its CC. However, the other two healers are much stronger at healing under pressure. SCH has free heals from Eos which are about equal to a ticking regen, and can be paired with whispering dawn. In addition to this, their Adloqium shields provide a bit of a buffer to get off more Adloqium casts to provide a constant stream of healing while getting pounded. To top that off, SCH has lustrates much more readily available than Tetragrammaton or Essential Dignity. Astrologian has Aspected Benefic in their arsenal, which is incredible for keeping yourself up while LOS-kiting back and forth around walls, or around pillars. Pair this with Essential Dignity, Lightspeed, and your choice of built in HOT stacking or shields? AST does a great job of surviving on their own.
WHM, without their CC, only have their HOTs and hard casts outside of a 5 minute benediction CD or a 1 minute Tetragrammaton CD. This is supplemented periodically by hard-cast kiting with assistance from POM, Equanimity, and Surecast.
As for your example of how badly WHM counters healers with fluid and sleep...Have you considered that you may be out of position and opening yourself up as an easy CC target? Any healer should be able to LOS enemy BLM/WHM to a great extent. And if you are getting chain slept with a fluid aura weaved in, that WHM is either blowing all their OGCDs to guarantee you are slept and that their team doesn't die...Or your team is not applying enough pressure which allows the WHM to "win more" and close out the game with intelligent CC's.