That said, I'll say that while technological tools are useful and important, making this kind of thing work is less about technology and more about people, culture, incentives, etc. What do I mean by that? I'm talking about things like, the way people in a given firm may be perversely incentivized to not share knowledge and help groom a central KB, because their current state of being the sole possessor of some bit of knowledge gives them (power|influence|job security|etc). And the way people exchanging knowledge in a firm are basically engaging in a sort of market exchange, where market dynamics apply. People "trade" with people they trust, and they engage in heuristics like satisficing, and favoring availability over veracity, etc.
So yeah, you can throw up an instance of MediaWiki pretty easily. And I do encourage it, as I am a fan of wiki's, even though they have issues (they quickly become full of outdated and irrelevant content, and become a disorganized clusterfuck without a lot of continual grooming), but don't expect having a wiki (or Sharepoint, or Alfresco, or $whatever) to be a cure-all.
I think that which technology to use (wiki, shared drive, etc) is not as important as the fact that there should be one AND ONLY ONE place where documentation lives. This is one of the reasons why I hate MS Teams: because every channel has its own Wiki plus its own Files, meaning that I have to hunt through every even-remotely-relevant channel and through several tabs before I find something I know exists. The same goes for chat - once half your team uses Slack and the other half prefers Skype, notifications will get lost.
Second, I think it's a good idea to have someone whose official job includes nagging other people about updating documentation rather than waiting for everyone to do it by themselves. I was in charge of keeping a log of weekly research meetings, and the simple act of writing
March 25: Presentation by Rudolf (Paper: ?? - Slides: ??)
on the designated wiki was a great mechanism for getting Rudolf to upload their slides.
That's the best method.