To get back at her husband. That and also Ryubee doesn't have the tech to make those types of memories. His Gaia memories are primitive and the only riders he could make were that of his own family. He had to negotiate a deal with Foundation X to lease their tech.
Espionage & Sabotage (to take it over). Since shroud has no need to hurt anybody else as Ryu said, she left Kazu in charge. She now has no control over Foundation X.
Also let me say: Narumi got the Lost Driver from
somebody and it wasn't Ryubee. Eternal (Foundation X tech) used a Lost Driver. Put 2 + 2 together.
Then you have to explain this:
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That device is connecting her to the earth itself. She's a cleric. We never really did see what Terror did to her, just the purple goo and it fast forward to a later scene.
Dude. You're just making so many assumptions here.
About Shroud - What makes you think that device allows her to connect to the Earth? From what we know for a fact, Terror used his goo on her face and injured her. Hence, that device could be anything. It could be a voice changer giving her the deep voice she has now; a life sustaining device because of her injury; a device to heal her wound; or something that turns her invisible, since she can come and disappear so suddenly. Heck, what I'm trying to say is that it could be anything within the bounds of logic, with consideration for Shroud's background and unique circumstances. However it is most likely NOT something to connect her to the Earth. The only ways we've seen this happen are for someone to die in that well (Raito) and be infused with the Gaia Progressor, or for someone to fuse with the Gaia Progressor (Wakana) at the molecular level. There has been no evidence that Shroud has been modified that way, nor is there evidence she can achieve the same thing with that device on her neck.
Also, I do not see how you're conceiving a connection between Shroud and Foundation X. All we've had so far is Kazu saying that the T2 memory research was started by someone from Museum (hinting at Shroud) and continued by them. Besides that, we've had nothing else to hint at a connection between the two. From what we've seen, Shroud is shown to be a lone warrior plotting behind the scenes to destroy Terror. Her only purpose is to ensure the safety of her son, so if she set up Foundation X to provide technology to Museum, as you've claimed, isn't she doing the exact opposite of what she wants?! If anything, Shroud and Foundation X are completely separate players in the conflict of the show with vastly differing motives.
Furthermore, Foundation X is NOT providing technology to Museum. All we know is that they provide funding for Museum's research; they're nothing more than sinister shareholders. I remember Kazu and Ryuubei meeting in an earlier episode with Kazu threatening to withdraw funding if Museum doesn't increase production of Gaia Memories. Hence one can infer that Foundation X invests in Museum with an eye on financial gains (and possibly other types of profits).
Also, I don't get how you find Museum's technology inferior to Shroud's. Ryuubei has the technology to initiate something as momentous as the Gaia Impact, and the ability to turn Wakana into her Xtreme form, so he's at least on par with Shroud. Plus, if you're gonna argue about the impure memories that he sells, in episode 46 he's stated that those memories were sold to people as experiments. Hence it can be inferred that he intended those memories to be raw, so that he could gather data on their effects.
One last thing. We still have no idea where Eternal's Lost Driver came from, nor do we have confirmation whether it was in the case with the T2 memories or whether he had it with him from the start. Hence, you cannot claim that the Lost Driver is Foundation X technology, and hence use that as proof that Shroud is connected to Foundation X.