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Another thing to point out, with the new patch. On a dedicated faith build with low str and dex investment, Blessed infusion is now stronger than uninfused, heavy, sharp, refined, and raw upgrade paths. At least on weapons that scale S in faith, which is quite a few from what I can tell. it seems all the weapons that previously scaled A, now scale S. The list can be found here:
Sadly that list is pretty outdated, in the patches since its likely at least some of those have changed. I think someone on reddit will probably put out a list soon.
Alright. Well, it may be outdated for some infusions or weapons. I only checked the Blessed infusions. And every weapon on that list that said it previously scaled A in Faith I checked and all of them now, as of patch 1.13 scale S in Faith when at +10 (most at +9). There may be even more now tho, as I only verified the ones on the list.
All of these weapons (and possibly more) scale S in Faith when at +10, and most at +9.

Lothric Knight Sword
Gotthard Twinswords
Astora Greatsword
Carthus Sword
Painting Guardian Curved Sword
Follower Sabre
Carthus Greatsword
Exile Greatsword
Harald Curved Greatsword
Richard's Rapier
Crow Quills
Onikri and Ubadachi
Hand Axe
Thrall Axe
Winged Spear
Rotten Ghru Spear
Four-Pronged Plow
Follower Javalin
Drang Twinspears
Lothric Knight Long Spear
Lothric War Banner
Great Scythe
Great Corvian Scythe
Oh, this is interesting. Lothric Knight Greatsword, when Blessed +10 A scaling in Faith, isshowing a 705 2h AR with 18 Str 18 Dex 60 Faith. From what I can tell, it looks like the Lightning Damage on the weapon when infused with Blessed scales off the Faith stat. That's from the mugenmonkey, as this build is not that far yet, so I can't see the actual numbers.

Oki I looked on my 55 Faith 35 Strength 18 Dex build. It would seem that the Lightning damage indeed scales with Faith. Here are the numbers.

Blessed Lothric Greatsword+10
Scaling D in STR, C in DEX, A in FAITH

Physical: 185 + 280 (2h 290) = 465 (475)
Lightning: 113 + 127 = 240
so 707 1h, 717 2h

Compared to the Lightning +10 with same stats:
Scaling D in STR, D in DEX, and A in FAITH
(the A is red which would seem that the Faith scaling is weaker than Blessed)

Physical: 157 + 55 (2h 62) = 212 (219)
Lightning 231 + 239 = 470
so 682 1h, 689 2h

So in conclusion, Blessed seems to be the better infusion on the LKGS. Tho the Lightning does considrable more Lightning damage.
With the latest patch blessed gem based regeneration tanks build is possible. Currently using Greatsword (UGS) infused with blessed gem on 28 Str , 10 dex and 39 faith, two handed it deals 512 (200 + 312) physical damage at +10. Not as high as what a quality build would get, but still strong enough to destroy enemies, coupled with high vitality and absorption, it becomes a very durable build with sun princess ring and blessed caestus.
Why no great swords with S scaling? Ultra greats looks stupid and have no sheath.