>>725032139
I would be careful of reading too much into that because they're Hornet's in-character observations and not necessarily an authoritative word-of-god. How's Hornet going to know if Phantom was ordered to operate the Exhaust Organ, or if she's going it because she's an emo teenager all up in her feelings. "I'm garbage this is where I belong"?
I will say that I don't believe there's any overlap between the time of the Weavers and the creation of the silkspun sisters. We know from Pious Isamor that the Weavers bequeathed the Citadel to the mortal bugs of Pharloom in a much more austere state than it ended up. The mechanical infrastructure to support the Cogwork Core all would have had to come much, much later, after the song has started to falter and the Architects were tasked with coming up with an automated alternative.
There's some (likely intentional) ambiguity, but I don't think that GMS started to awaken until after that, and Phantom and Lace being essentially messianic figures of their god-mother's impending return would not have been permitted to exist if the Weavers were still around.
Most likely sequence is, Weavers overthrow GMS, keep her asleep themselves for a time, foist it off on the mortals by establishing the Citadel of Song, lurk in Pharloom for a time while they try to figure out how to permanently destroy GMS, then flee from Pharloom when they can't.
The Conductors, meanwhile, become decadent as the heads of a Weaver-worshipping cult, then complacent as they get comfortable with a constant influx of pilgrims to raise them up. As pilgrimage begins to decline, they resort to increasing reliance on automata - first the Sentinels to abduct bugs to serve, then the creation of the Cogwork Core itself to power the autochoir - and silk suture to keep a dwindling number of choristers singing longer. The order of operations here is a little ambiguous but may overlap or occur concurrently.