Thanks for the feedback! To address some of your concerns:
The being wants to break through into our universe because it consists of the entirety of its own universe. There is nothing there that isn't it, no void, no planets, just an infinite dimension of this one, impossible entity. The long and short of it, is that it's bored. Alone, lonely, unable to do anything other than just be. Trapped by the very nature of what it is. Once it realized that it could get through into another universe, that's all that it wants to do. Desperately wanting to go somewhere where it isn't the only thing in existence.
The people that it creates are all literal extensions of itself. Closer to appendages than separate individuals, though they are capable of observation, conversation, ambulation, all that good stuff. The Foundation doesn't realize at first what the connection is, so much as they notice that researchers on staff seem to be accessing records, SCPs facilities, etc. at an alarming rate, that they shouldn't really be able to do. Such as one researcher's credentials being used in a facility in one location, and then at a facility in another location hundreds of miles away only a few minutes later. The implication being that the entity is supplanting actual researchers, getting rid of them and taking their place. Though now that you mention it, I really like the idea that it does create infants, possibly replacing babies and letting them grow naturally over the course of decades, in order to gain as much experience as possible. Perhaps it can do both?
As far as the obvious issue of people noticing it disappearing, that's part of how it gets noticed by the foundation in the first place. The entity isn't used to subterfuge, or even really thinking about how other people think, as it went its entire existence being the only thing in existence. It's not necessarily good at it, and is prone to making mistakes, though it gets better at it over time.
The nature of the entity becomes made clear through a series of interview logs, culminating in the final log, where it attempts to strike a deal with the Foundation: it gives some explanation of what it is and how it works, explaining that it can infiltrate other GOIs to get information for the Foundation, in return for gaining access to information that it feels would help it for its own goal. While also dropping the not-so-subtle threat that if the Foundation doesn't help it, it may well approach those other GOIs with the same offer. Leaving the Foundation with a difficult time trying to figure out just how to deal with a potentially massive information leak, whilst also not knowing just how many people in their organization may actually be probes of this extra-dimensional entity.
