@SmilerLoki, I included those quotes already. 
I know, I just added the link.
I have no idea how you conclude that Gods with Alzeihmers can do anything.
The same way that people with it do. They forget
some things, but not
all things. Their memory is not erased completely and instantaneously. By analogy, the Gods are uncomprehending to some things now, but not to all things. What they still know, they can interact with.
We're talking about an inability to grasp or form memories in creatures that depend entirely on the memories they harvest from souls.
We have no evidence of the Gods needing souls to have memory. We still have no idea whether the Gods are dependent on humans/souls, and if so, then why. They don't seem to be D&D types of divinities, or at least not even close to fully. And if the Gods
are dependent on humans, then if their connection to the world is severed as you postulate, then they should cease to exist. I do think it's what the No-God is for, but I don't think it's accomplished. I think it only really starts when the population is cut down to the 144k.
No more babies = no more souls. Maybe they can peer uncomprehendingly through birds or whatnot, but I don't believe the Gods can intercede from the outside.
There are still living people, who have souls, even working under your assumptions.
Basically, saying the Gods cannot intercede means the Outside cannot intercede, which would mean it's shut off. It isn't. It's impeded in some ways, mainly concerning the Gods, but not at all in others (at least we have no evidence of that), like sorcery.
It could be that Skafra is lying, but I don't really find any reason to think he is, plus the further anecdotal evidence point to the fact of "souls that encounter the No-God pass no further" as being true.
I also don't think he was lying here. What he easily could've been, though, is mistaken (completely or partially). As creatures of the Inchoroi, dragons seem to be heavily indoctrinated. And their "religion" had already been ill-understood by the Inchoroi when they created Wracu.
That being said, it's clear to me (since I find textual evidence of it) that the No-God indeed interacts with souls. But the extent, method, and end-result of this interaction are all completely unclear. And not in a "I can't imagine what happens here"-kind of way. It's much worse, actually, since I can concoct dozens of workable theories. The problem with all of them is the same: lack of information. Right now the effects of the No-God are defined so vaguely they can be interpreted in contradictory ways. And there is no opportunity to test theories. So when a need to explain something in one specific way arises, it basically means there is no possibility to consider that theory true. It becomes just one of many equally probable speculative outlooks.