Okay, so I haven't read 3916 before, and if I have, I've forgotten about it. So I'll essentially be representing that portion of the audience, I guess!
Does this read like it would work as a standalone piece, or should I make it more of a supplementary-article-type thing?
It does read like a standalone piece (in that you don't need to have any specific prior knowledge to understand what's going on), but seems to be more of a prologue than a full-on story.
How's the length/pacing? Do any of the sections feel too long or too short?
Length seems okay, pacing is a little bit on the unsure side for me since again, seems more like a prologue. Nothing really happens except for a recap. If that's the intention, then you might want to intersperse the recording with some actual plotline from, say, second person perspective of whoever is currently listening to the recording. Then you can have some sense of progression of events while the reader surrogate works their way towards the Foundation and against the locusts and whatnot.
I will say that the recording kind of gives away more than it needs to with regards to the Foundation. I don't think the anomaly explanation ("stuff like immortal omnicidal lizards, reality benders, and sentient ideas") would make much sense to someone with no clue about the Foundation or anything anomalous. SCP wiki readers might already know what a reality bender is, but a completely clueless civilian would probably just be told "dangerous things that science can't explain" or the like.
Also, the "And the answer is because we couldn't. That's not who we are as an organization. We secure, contain, and protect not just mankind, but anomalies as well." answer feels a little wishy-washy, especially since the Foundation could have just frozen the locusts or kept the populations down before the flesh-stripping incident. (But then, that's just a quibble from me personally, and I headcanon the Foundation as extremely competent. Plenty of others may very well happily accept the Foundation's slip-up here.)
How's the ending? I feel like it's lacking a little something but I can't quite figure out what.
The ending comes off as overdramatic for me, especially with the "Secure, contain, protect." sign off (which effectively would mean absolutely nothing to a civilian listener, especially if they didn't really comprehend the meaning of the Foundation motto when it was introduced earlier). I feel like it would make more sense to just recap the entire thing in very simple terms: the locusts are the Foundation's fault, but they left a failsafe. It's in Yellowstone. The listener needs to get there and resets the world. Then everything will be fine. Promise. Please do this for humanity. End scene.
