...where do I even begin *Rubs temples*
You are mistaken. Spell speed reduces the cast time and recast time of your spells. All Lightspeed does is reduce the cast time of spells by 2.5s while it's active. You're not using GCDs any faster (though you are front loading your heals). That's just an affect of your Spell Speed on your GCD when combined with the cast time reduction of Lightspeed (maybe a small exception with Gravity but you shouldn't be using Gravity with Lightspeed normally).
Any potency AST can get from SpS, WHM can get stacking an equal amount of SpS. I'm not quite sure what's the point of this comment.
This isn't proof that AST has better healing than WHM. This is just proof that maybe you reacted to the failed mechanic much better than the other healers did in the other alliances - thus you succeed and they wiped. If you look at the actual potency per second of each spell and potential buffs at level 50, you get this order of magnitude:
Medica = 300 Potency @ 2.5s cast time = 120 potency per second per person
Helios = 300 Potency @ 2.5s * 0.95 = 126.3 potency per second per person
Medica w/ PoM = 300 potency @ 2.5s * 0.8 = 150 Potency per second per person
Helios w/ Synastry = 300 Potency * 1.2 @ 2.5s * 0.95 = 151.6 potency per second per person
Medica w/ Divine Seal = 300 Potency * 1.3 @ 2.5s = 156 potency per second per person
Helios w/ Synastry + Arrow = 300 Potency * 1.2 @ 2.5s * 0.95 * 0.9 = 169.0 potency per second per person
Helios w/ Synastry + Enhanced Arrow = 300 Potency * 1.2 @ 2.5s * 0.95 * 0.85 = 179.10 potency per second per person
Medica w/ Divine Seal + PoM = 300 Potency * 1.3 @ 2.5s * 0.8 = 195 potency per second per person
Also not sure what you mean by "not eating up Mana" in your post since AST doesn't have an AoE heal ability that isn't a HoT.
Every job has their nuances that require them to play well and have different tactics and tricks that work well with their kits. Divine Seal + Regen + Medica II + Asylum (360 potency / tick) pretty much equals AST's super HoT ticks (366 potency / tick) at the cost of having less duration but WHM "mitigation" can be extended on trash via Holy stun on Trash for an additional 7 seconds. The most obvious trick in the SCH kit is Deployed Adlo, but can combine with Fey Illuminate + Dissipation for insane mitigation in a single sitting if required.
Yes, it's annoying when people say that AST can't compete as that shows a level of ignorance of a past incarnation of the job. With that being said, telling everyone else to "learn to play" that you are prone to do in a myriad amount of your posts isn't the way to do it. Instead it just shows a level of ignorance from you about the entire healing paradigm. Right now, all I get from your posts is that (1) you're an AST fanboy/girl (2) you show a level of contempt for other healers that's unhealthy and (3) you have a level of ignorance about game mechanics and it hurts your posts and logic more than it helps.