Playing a Healer reactively is quite possibly the worst way to play a Healer. It leads to second-guessing, indecision, not planning your resource expenditure, and plenty more things that will make healing the fight harder, and to top it all off it will lower the damage you do as a knock-on effect of all of these.
While Scholar and Astrologian have more in the way of shields than White Mage, that is not to say that White Mage should not be being just as proactive.
Have you ever played The Chrysalis in i90 gear on White Mage? This was the trial that really hammered this home for me, back in early Heavensward when I first started playing. Double and Triple Spark were doing so much damage to the Tank that they would die almost immediately, and so I had to learn the spells' timing and cast Cure II such that it would hit the Tank after the first Spark, but before the second, and then later when Triple arrived I had to do this twice back-to-back.
Had I been reacting to the Tank taking damage, I would simply not have had enough time to cast before the second/third Spark hit and deleted them, and we would wipe. By quickly learning when in the spells' cast I could begin my own to hit during the delay, I learned to act proactively, allowing me to keep the Tank alive.
Additionally, I'd like to mention that with the removal of Cleric Stance, Healers no longer have a 'DPS mode' that they're locked into, any such phenomenon is entirely a player-sided problem, and one I suspect born from this kind of thinking. Abandon the concept of having two 'modes' and fuse them together. Any time you don't need to be casting a healing spell, cast a damaging one - you can always cancel out of it if you realise you actually need to be doing something else; three-and-a-half Stones and a Cure II is better than zero Stones and a Cure II, after all.
As I touched on a little previously, don't think of it as a 'DPS phase' and a 'Healing phase', but rather think of it as 'Healer gameplay', all in one soup. If you wait for windows in which nothing is happening to deal damage, you will end the fight with no damage done. Find the points in the fight that you need to be healing or doing mechanics, and work backward from there - any point at which you're not healing and not doing mechanics is a point you should be dealing damage.