The thing is there's a difference between having the data and actually doing the analysis. Think of it this way: Survey has a library, with all of the information she has stored in books in that library. Information that she doesn't have the permission to do any analysis with is the equivalent of it being added to the index for easy referral later on, while actually doing the analysis involves reading the books. If Survey already was doing the analysis for everything she had and merely withholding the results, then 1) she would've immediately pointed out Theo's relationship with Kaiser and 2) it would've also gone against the whole AI development thing where she slowly learned to prioritize and focus on salient details to some degree instead of just blindly analysing every avenue available to her. Sure, it's in her capabilities now, but Survey's development in this fic has largely been in the aspect of not being focused on the big picture all the time.
I sort of get what you're going for, but I'm not sure it's
possible for Survey to gather information without analysing it at least a little. If it's a whole database she can simply copy down to one of the Workshop computers, sure. That would be like buying a book, taking it home and putting it in a book case without ever opening it.
But more complex and scattered information, like about the individuals surrounding Joe, that would be more like... Hiring a research librarian to find the information? They would need to be given the task, search for and pick out many different books in a huge library, then read and find the specific pages or paragraphs containing the information needed, copying it down, before looking up
new books that are refereed to in the one they just used, and so on. It's a far more involved form of search, and even if it's not
their research and they don't
try to look for any meaning, just the fact that they have to read and search for things means that a certain degree of analysis is done and conclusions are drawn no matter what.
The fact that Survey hasn't told anyone that Theo is Kaiser's son doesn't really prove anything one way or the other. Either Survey
doesn't know, and hasn't said anything because of this. Or she
does know, but this type of information is both private
and falls under the umbrella of the Unwritten Rules about not unmasking or trying to find out the civilian identity of capes, which is something Joe has stated he intend to follow. Which means that Survey wouldn't say anything because of this. Just like she hasn't mentioned that Grue and Lethe are siblings to anyone, even if she has known that for a long time now.
As for the AI development, I agree that a big part of her development as a person and character has been to learn to limit and prioritize her data search and analysis, but that would have the
opposite result of what you describe. She started out with wanting
all the data and doing
all the analysis -almost to the point of madness- but as she has grown she has, painfully, learned to sort and prioritize what to do.
This means that even if the Survey of today would be more restrictive with how much information she gathers and the analysis she does,
younger Survey would still have been in the '
all the data
all the analysis' stage when she first was allowed to interact with the outside world and do her first forays of information gathering. Finding out
everything she could about the people directly interacting with Joe would very likely have been a high priority for her.
The older and more mature Survey would perhaps not do this, but her younger self did, and she can hardly
forget the information she's already gathered and the analysis already done. (Or, well, she
could. Because she's an AI and presumably has the ability to edit her memory files if she want, but this seems like a bad idea in so many ways. Especially when she simply can put the knowledge to the side and label it as 'confidential' instead.)