Because, of course, Contessa is Cauldron's only information-gathering option. It's not like they have a Cape who can observe every location on Earth at will.
The Clairvoyant can, as far as they know, see every location on every Earth that the entities both have access to and have not blocked Parahumans from accessing, hear anything going on at those locations, and share that view with anyone that he is currently touching. However, he lacks the ability to express what he sees/hears, and is basically limited to being the "eyes" while the Doormaker is the "processor" and the "door". Even should another person come in contact with The Clairvoyant, they will be unable to express the information they learn while connected, and, when disconnected, will suffer severe backlash including a short coma and memory loss...memories such as what, specifically, the saw.
It doesn't sound like either of those would apply to "precogging" the present like by asking PtV the name of the person creating the technology. You don't need any outside data to do that, just existing on Earth data, and you don't need to understand the phenomena to know the name of the person who typed it out.
The PTV does not GIVE an answer. Contessa cannot ask "who made this", and get a response. She has to ask "Path to knowing who made this", and then, USING PRECOG, the PTV gives instructions one step at a time (and the number of steps required) to know that information. She doesn't use this option most of the time, because for a lot of paths it ends up being "Kidnap this family, extort this relative for information", perhaps with "kill the kid to show you are serious" included. That is why they have put Alexandria as the Chief Director, and have Number Man dealing with basically all the money: So they have options other than someone getting Contessaed.
The basis for Precog does, of course, get updated from all the intelligence-gathering shards, from shards like Tattletale's that process information as their main purpose, to shards with secondary sensory powers like Gallant's emotion sense. But that basis gets cropped for all data related to the Entities and the Endbringers, then gets pushed out to the precog shards who put together a whole picture, which then gets cropped AGAIN for everything related to blind-spots, and then the precog shards give information to their hosts in specific, and often hard-to-understand, ways.
As to the "Cauldron has five people" thing, the only "sensible" way to fix it is to accept that those are the five HEADS of Cauldron. Like, Rebecca's secretary is in-the-know about the double-life and the fact that Rebecca literally goes away, as is Rebecca's body-double (which may actually be her secretary as well) that appears when Alexandria needs to be in a video call with the Chief Director, or vice versa. There are likely a lot of people working for the Number Man that help manage the economies of the worlds, just like there are probably multiple people that have been tapped to work for Contessa.