Note that Star Trek faces potential liability from allowing fan film makers to assert their copyrights. That's got to be one reason for the decision to discourage it.
Example: Suppose Star Trek Discovery makes an episode about a character who dies, but is "resurrected" as an advanced hologram of himself, and has to come to terms with that.
The morning after it airs, Eric Busby files a lawsuit against CBS/Paramount alleging copyright infringement of "The Sum of One's Memories," episode 2 of The Section 31 Files from Darker Projects. (Busby would never in a million years do this, and especially not now that he has distanced DP from fan works -- I am only using him as an example.) Now, I'm willing to bet darseks to doughnuts that only a tiny fraction of the people in this thread have ever heard of The Section 31 Files, much less listened to "The Sum of One's Memories" (my favorite episode of theirs, for what it's worth). And, if a bunch of hardcore nerds like you haven't heard of it, how many people at CBS have?
Yet Busby does own the copyright on all original elements* in "The Sum of One's Memories," because that's how the Berne Convention works. So he could conceivably have just enough grounds to litigate. (He'd lose in the end, because he'd never be able to prove CBS had prior knowledge of his work, but the litigation would be costly.) And that would be an enormous headache for CBS.
This is one of the reasons, incidentally, why Star Trek eventually had to shut down its open script submission program, though they valiantly kept it open for years after the lawyers told them to close it. A freelance writer would submit a script; the writing staff on Star Trek -- without ever having any contact with the open submission department -- would make an episode with similar elements; the freelance writer would decide Paramount stole his idea and sue; Paramount would invariably make the problem go away, but at no small expense.
I always thought Trek should bring back the submission program, but with the stipulation that you forfeit all rights to sue Paramount over anything related to it if they don't purchase it.
Anyway, I'm going off-topic. Point is, there's a strong CYA element at play in the provisions against copyright assertion. I can't blame CBS for including it.
*Just the original elements, not the Star Trek elements. Separating the two is a court case in and of itself, but suffice to say that there are some copyrightable and therefore copyrighted elements in any episode of unlicensed fan fiction, which, under current IP law, are owned by the author, not the IP holder.
A point of clarification:
Eric Busby has nothing to do with Darker Projects these days. Though he founded it, it was mutually agreed upon by the people responsible for keeping Darker Projects running that he part ways from that group. Following several ugly episodes wherein he treated most of the volunteers who wanted Darker Projects to expand into original content over fan fiction, he exited with his Angel/Doctor Who mishmash "The Byron Chronicles" and a few of his other audio dramas about three years ago in favor of his own startup/site, EricBusbyPresents.com after numerous people at Darker Projects had had enough of his poor leadership and glory-hogging.
Voice actor Mark Bruzee took over after being voted in, and now MJ Cogburn runs what is left of Darker Projects.
That said, the rest of your post is spot-on. "The Sum of One's Memories" is/was a fun, clever episode of The Section 31 Files. Busby may talk a big game on his Facebook page, but as of at least three months ago he was no longer a member of Darker Projects.