After much discussion and learning from an experienced healer who has "cleared hard content", I am finally enlightened and would like to share with the world what I have learned and why DPSing as a healer is against Yoshi P's desires and hence a sin.
A) You can't spell HEALER without "HEAL". But no matter how you rearrange the letters, you can't spell "DPS" from "HEALER". There is no D (damage), P (per), or S (second) in the word "healer". Clearly, this is intended by Yoshi P as a means to subtlely tell us healers should heal (since you can spell "heal" from the letters in "healer"), but not do damage (since you can't spell "DPS" from "healer").
B) The game explicitly encourages you to stand in AoEs by example:
Clearly, players are NOT meant to avoid AoEs. They are meant to take the damage and you as a healer need to heal through it. Otherwise, the game won't let NPCs in Hall of the Novice stand in AoEs. If everyone stood in AoEs, the way the game is meant to be played as Yoshi P intended, then healers will not have time to do DPS. Healers are only doing DPS because people are playing the game wrongly.
C) Ad Hominem attacks on healers who don't DPS
As evidence from several other threads, people would rather try an insult and bully healers who don't DPS than actually consider the idea that there is another way to play, and that is what Yoshi P encourages. Because of all the ad hominem attacks going around, people forget to try to play the game the way Yoshi P intended.
D) Using HoTs only is lazy
Healers who DPS mostly only rely on HoT (healing over time) spells. These healers claim they are only healing 17% of the time... Now, if you're healing only 17% of the time, and healing is your main job... you're only working 17% of the time. That is like going to the office as a programmer, and only programming 17% of the time. And you spend 83% of your time watching videos. This is what healers who DPS are like. They heal 17% of the time and spend 83% of their time NOT healing. Tell me who is the lazy healer here.
E) The dungeons have a 90 minutes time limit
Yoshi P said he designed the game with NO healer DPS in mind, which is why dungeons have a 90 minutes time limit. If Yoshi P encourages healer DPS, then why would dungeons have a 90 minutes time limit? Right now there are too many speedrunning groups, with healers doing DPS and being lazy and not healing. They are not playing the game the way Yoshi P intended. If it was intended by Yoshi P for healers to DPS, then we will see dungeons have a shorter time limit closer to the clear times the speedrunning groups achieve. But right now, it's at 90 minutes, and speedrunning groups are only spending 15% of the time limit in a dungeon, which is completely against how Yoshi P designed the game.
F) Healers play healers because they want to heal. They don't want to DPS.
G) Your heals are very bad in cleric stance. If Yoshi P intended for healers to do damage, then why would your heals be very bad when in cleric stance? Cleric stance is meant for healers to do damage when they are alone doing FATEs or quests. They have their chocobos to heal them so they're a DPS. In dungeons you can't pull that off because if you go into cleric stance, you can't heal and people will die. If a healer is using Cleric Stance in a dungeon that means they're being lazy and will cause a wipe since their heals won't be high enough to counteract the incoming damage.
H) The White Mage "lily" mechanic is a sign from Yoshi P to tell healers to start healing more. Why isn't the "lily" mechanic tied with damage? It's tied with healing because Yoshi P knows the NA community is too toxic and healers don't want to be healers (which is why you have healers healing only 17% of the time). Yoshi P is telling us, with the lily mechanic, to start healing again otherwise you'll never get lilies.