Yes, and Accuracy to cap is the most appropriate way to stat your gear as a healer in FFXIV, because, again, it's the most gain per stat point in your overall performance, of which DPS is an integral part. The other stat points barely increase your healing throughput, and I generally go by the simple rule that if you cannot heal an encounter without the minuscule gains from melded substats and your weapon, you will not be able to heal it with those substats either (and vice versa). Accuracy is the objectively better stat to prioritize on, no matter which healer you play.
I don't mean to sound rude, but another healer once told me this simple truth: if you don't need extra PIE on AST or WHM, you simply do not DPS enough, and it's generally true. That extra PIE is also about sometimes being able to choose GCD efficiency over MP efficiency (such as using Cure II instead of spamming Cure I to get back into Cleric earlier), making room for fuck-ups during progression, and other more fuzzy benefits. Also, specifically for the relic, if you find you don't need any of the extra PIE, you can always reallocate it to a substat of your choice until you are at a comfortable level.Only when shit hits the fan do I run low on MP, and that's from spamming heals. Yes, I do help push DPS whenever I'm able to. Standing around and doing nothing is unproductive otherwise. If you pop your LA on CD and maybe supplement with a Ewer if you feel you may need it, you shouldn't have to worry all that much about MP.
Also, as I've said before, AST MP solvency is only marginally better than WHM in theory, and really only pulls ahead if you consistently use the 10th LA tick. This comparison even ignores a lot of the other aspects of the two healer's toolkits, and I'm pretty positive Freecure procs alone level the field if you properly utilize them, but I'm not a theorycrafter. I can search for the appropriate reddit thread if you want to.
And your claim is wrong. Spellspeed can have very noticeable effects on your timings on boss fights. For instance, on Nidex with 800-ish SpS (2.33s recast), I can pretty consistently fit in two Stone IIIs plus an Aero before Eggnogg takes off for the first dive, with less SpS, I can only fit in one Stone III plus an Aero. With even less SpS, I would not be able to fit in a Stone III after my opener before I need to switch out of Cleric for a bit to heal the tank and group without messing up all my timings. And yes, more SpS can easily allow you to better dodge things or heal between movement mechanics, you don't need to be a "super eSports player" for this. I agree that SpS can be fuzzy, non-obvious, and subjective in its benefits, but it's false to claim you can't tell any differences. In my opinion, you should have, at the very least, enough SpS to consistently fit in another spell into your buffs, such as Divine Seal, Presence of Mind, or Synastry (which is the case at around 500-600, depending on latency).