I agree that it would be preferable if only subject matter experts are allowed to contribute to the docs… the whole "in an ideal world" and so on.
However, I am not sure limiting by badges is the right way forward.
Regarding "rep farming":
I am a member for 5+ years and seen a fair share of gold badgers rep farm, including me. There is no guarantee people will not use docs "for teh reps". Having an entry barrier will obviously lower that amount, even if it just meant that the rich get richer, but it will not prevent it altogether. You never claimed that though to be fair.
Regarding contributions:
You say
The main reason experts don't want to contribute to documentation because at it's current state documentation is flat out crap. Not just the UX but also the content itself.
Even if they rebooted the docs, fixed the bugs, UX and structural shortcomings and allowed only badgers to contribute, I cannot be bothered to contribute. I just don't see utility of that whole project. I don't see why I should replicate a good portion of my 2602 answers on Q&A over there. It feels wrong. I fought years against duplicates on Q&A and now I am invited to duplicate my stuff there? Why?
As I have pointed elsewhere on MSO, the scope of this docs project is unclear to me. It still is. Replicating my answers cannot be its purpose. Duplicating existing official documentations cannot be its purpose either. And no one needs a bucket of random examples. If you want to contribute a random example, just provide them as self-answered Q&A.
IMO, Docs is competing with Q&A and official documentations for no good reason. The idea to provide docs for otherwise poorly documented projects is laudable, but The One True SO Documentation is a wasteful and arrogant effort. Stack Overflow should not attempt to replace an authoritative documentation made by project maintainers elsewhere on the web. I will not participate in that.
The beauty of Stack Overflow for me always was that it gives me quality Q&A and links to authoritative outside sources for further digging. As such, it was a peaceful coexistence. I'd like that to continue. I don't want a Stack Overflow silo, where every answer links to some SO.docs instead of official docs. And I don't want the first ten pages of Google SERPs to be to SO only. Diversity is good and needed. SO docs is not.