Healer's will always be a necessity because no one can recover damage as consistently and strongly as one with that green icon. The thing you have to consider is that the healing output these healers can use usually far exceed the amount of damage a pack of monsters / boss may inflict in a single auto attack round / ability. This leaves a window of opportunity for healers to use the DPS part of their kit to help supplement the damage of the party and just make things faster. A dead mob can't hurt your tank anymore. DPSing is indirectly reducing your HPS requirement for the fight by causing things to die faster.
With that being said, DPS on a healer isn't a necessity to clearing content short of Savage. But once you hit a certain gear and/or skill threshold, you won't need to use your cure spells except for perhaps once ever 10 seconds or so - allowing a skilled healer to add more to their party via DPS.
However, if you're looking for a sense of urgency, you might not find that very often in this game short of Savage runs or undergeared tanks pulling much more than they should. You aren't necessarily another DPS who occasionally heals - you're a powerful healer who can supplement your party with subpar DPS (though if you do manage to match anyone's DPS on AST without overly outgearing them, shame on them..). A party won't function without that green icon there so a healer is still a necessary backbone.
Outside of a select few fights *Looks at Bismark duing P2 / final phase A4*, HPS requirement in this game is often very light. It's easy to get away with a solo heal in most non-Savage content and two skilled healers weaving stance dancing in their healing can easily do 800 to 1,200 DPS combined in most trials / Alex Normal.
You can bypass the character limit by editting your post. I'd recommend typing it out then cutting out the bits that exceed then 1,000 limit than edit and paste the words back. It's bit archaic but oh well.
In terms of your opportunities to DPS, as a WHM you get one of the best tools to DPS trash with - Holy. Pop Asylum + Regen or Divine Seal Regen on your tank and then get in Cleric's and hit Holy >> Aero III >> Holy >> Holy. This will give you 7 seconds of stun and then you can resume healing, while providing some DPS to your group. Just be aware of what mobs may be immune to stun so you don't put yourself in a bad spot!
Also, sometimes the best thing to do is just Cleric's than hit Aero III + Aero / Aero II then get out and keep healing. It takes practice and comfort and not everyone will be able to get it instantly. Still, if you want to improve as a healer, I highly recommend practicing and see what limits you can and cannot hit. Make sure to take into account your tank's gear and what cooldowns they're using to help YOU in the healing department.
[EDIT] Just seeing some of the posts your putting up - in some cases the only thing you can do is healing. The best healers are those who can adapt to the situation and make the best decisions for their party. As a healer, your priority is always the party's health first. Once that's at a comfortable level, then you can bring the pain to your enemies. If that comfort level is never achieved to a myriad of inept play from the party, then so be it - grit your teeth and heal on.