Fascinating discussion.
@Niara, as always, I'm impressed as always by the depth your reading and the clarity of your expression. But, having said that, I'm going to be a difficult rabbit and express some differences of opinion.
The Weave exists as something like a cheesecloth - between people who would use power, and the raw uncontrolled power itself. The Weave is what spellcasters interact with in order to effect change in the world and draw power into effect; shaping the weave in an area or in particular way manipulates the raw magic of the space in particular, more controlled, ways.
While this a good interpretation of some ambiguous source material, I see it differently. We're really dealing with the nature of weave. What, exactly, is the weave? The weave is the both the source of magic power in the realms and it's Mystra herself. Mystra is a special case among the gods because she is in charge of the weave she is the weave itself; she's different - perhaps unique - in that she is simultaneously holds the portfolio and is the portfolio. [Sword Coast Adventurers Guide p. 19] Just as theologians in our world ponder how a being can be both word and flesh we can imagine that the Mystran theologians concern themselves how their god can the being keeps and cares for the weave and the thing that is kept and cared for.
When a beholder uses an eye ray or a dragon breathes fire they are drawing on their intrinsic connection to the weave to do so [SCAG, p 19]. In 2nd & 3rd edition it's pretty clear that all magic was this way - all magic is Mystra and Mystra is all magic - Mystra could even cut off the gods from the weave if she chose to do so. With 4e and the Murder of Mystra the weave was destroyed. Upon her death it was 'revealed' that there was a source of power other than Mystra.
But it's not clear what falls into that category of Mystra-independent power or if those other pools of power
still exist after the refashioning of the tablets of fate. Because we can see weave as a cheese cloth - straining and purifying magic from a raw pool of power. But I think it's better to stick to the original metaphor: the weave is like a tapestry on a loom. When the tablets of fate were broken, when Mystra died, the weave was unwoven. But the weave was rewoven when the tablets were refashioned. When Mystra was dormant most 4e magic users were pulling on the raw stuff of magic while wild magic users were drawing from the tangled skein of dead Mystra's weave. But what is true now? I guess if I'm going to follow this metaphor off the cliff I should call the raw stuff of magic uncombed wool and ask: is there any raw wool left now that the weave has been rewoven? Is there a boiling pot of milk waiting to be strained through a cloth? I suspect the answer is: no, all raw magic was made into the weave once Mystra came back to life, it's curds all the way down. Sorcerers aren't tapping raw magic, they aren't dealing with uncombed wool, they are instead taking threads from Mystra's weave and braiding them how they see fit. If there is some Mystra-independent source I would suspect that it would the source that Warlocks draw upon. (of course in your own game, etc, etc.)
On Warlocks. I don't have the source at my finger tips but in one of the youtube 5e Q&A sessions it was said that Warlocks cannot fall - which surprised me. Even if your patron disowns you, even if you kill them they have opened your eyes to a source of power that you can continue to draw upon - the patron only helps warlocks to "forge their path ". So it isn't that case that the patron is weaving for the adventurer as much as they have caused the scales to fall from the eyes of the mortal. This is why your patron can be level 5 while you are level 20.
Now I do think your description of clerics is spot on. The gods do the weaving for the cleric and hand their followers pre-woven, off the rack spells. Upset your god and you are in trouble . . .
Now I absolutely agree with you @Niara that 5e really, really emphasizes the anything goes aspect of the game. If the DM thinks it sounds fun, it's consistent with the spirit of the rule - the rules are not there to restrict your fun. At the same time I think @Leucrotta has a good point - if things become too fuzzy then things lose
all definition. (which I believe was your point with the changes to the DnD species) Instead of seeing the realms as a clay that can take a thousand shapes you have a formless lump of mud. And let's not forget context of 5e DnD - 4e really screwed up the realms and, even a decade on, we're still trying to repair that damage. One of the defining features of Faerun is the importance of the gods in people's ever day lives - Golorian style atheists don't exist, atheists = the faithless / Greyhawk-style clerics of abstract goodness don't exist, some god is answering your prayers even if they chose to hide their face. No one wants to limit what happens at individuals tables but, at the same time, I want the realms to retain its defining features.
Here's my suggestion for squaring the circle . We know that Paladin's power comes from their commitment to their ideals. We know that in Faerun all power - even sorcerous power - comes from the gods. So a power arising from a commitment to justice
is still power granted by the gods.
Each oath should be overseen by a consortium of gods and some of the aspects of that oath should be spelled out: never attack without first issuing a challenge, always act to save innocent lives, never kill a defenseless animal. This is lore consistent - Tyr, Illmater and Torm are a lawful good consortium, allowing Paladins of devotion to choose one of the three would add something to the game. This is less clear in the lore but I always thought that Silvanus was the fatherly aspect of nature and Chantea the motherly aspect - they could be listed as the guardians of the Oath of Ancients.
Otherwise Faerun's paladins are little different than the formless mush that was the PoE paladin.