Here's what I think might have happened. I think this would explain pretty much all of it:
This whole episode began with an exercise involving new pilots, possibly that mentioned above. Those pilots were super excited, like "dude, what is that bird?!" excited. They were not trained observers, let alone allowed to fly over cities with weapons (as Elizondo has stated), just being trained to become those.
During one of the training flights they were practicing to use their instruments, like the ATFLIR and autotrack. They were just pointing those to whatever targets they happened to find from the distance, and they didn't have the necessary experience to identify mundane things or estimate altitudes, speeds and such things. They were just getting to grips with their own speed, as quoted from above:
“It’s pretty mind blowing,” Adams said. “Really up at altitude when you’re fast you don’t’ have the ground rush, but the lower you get the faster stuff gets moving. “
Since they were in the training, their performance was recorded, and the videos were analyzed. Someone wrote down the analysis, possibly with pilot interviews as well. The videos as such weren't classified, they didn't contain anything worth classifying, but there was that sensitive personal data of the pilots and their performance, so those were stored in a classified system because of that, not their content. Just like Elizondo said, the videos were not classified, but the system containing it was.
The descriptions of those videos mentioned the events like they happened, or how the pilots described them. They mentioned that they spotted some targets they couldn't identify. The words that were used might have even used the now familiar terms of unidentified aerial phenomena or stuff like that. The pilots may have thought the object was flying very fast and low, and it was written like that. That may be the origin of the supposed additional information from the pilots Garry Nolan for example might have heard through some chain of people.
Then we have the AATIP, or what was left of it in 2015. As far as the DoD was concerned, that program didn't even exist anymore, and they didn't have funding. So they weren't really in the position of performing proper investigations, interviewing pilots and so on. They were just trying to continue doing their then more personal mission. They were scouring the available databases with keywords, which happened to hit to those descriptions in that training video. They heard how the pilots were all excited, so they became excited as well, and perhaps just trusted what the comments stated.
Since that was just an unclassified training video, publishing it wasn't a problem, as long as all that personal data was removed. Easiest thing was to just drop it all, so there wasn't much need to properly analyze what can be made public and what not. All they had was an unclassified training video, and they just passed that through some of the necessary processes to make sure it doesn't cause them problems if they publish that. But it might be that it wasn't quite by the book all the way. It might be that they clipped those parts of the video that interested them back then already, as it would be easier to have someone evaluate just a minute of video than all of it.
In reality it's hardly classified information if some training video was taken during a specific exercise, especially if information from it was handled like this at the time:
Leading the charge in community involvement was 2nd Lt. Patrick Gargan, 6th AMW media operation Officer, who organized multiple media days and the release of up-to-the-minute information on the exercise's status. Within the short duration of the exercise, Gargan's efforts lead to over 100,000 FaceBook views, 11 local media stories, and 22 photos for the Department of Defense.
TTSA might just claim it's classified, as they don't want to reveal how mundane all this really is.
As a reminder, according to Leslie Kean, this video, like the others, was "cleared for release in August", so about 2 months before TTSA was announced.
Also remember that, according to Washington Post, these were supposed to be the "most unusual videos":
Just before leaving his Defense Department job two months ago, intelligence officer Luis Elizondo quietly arranged to secure the release of three of the most unusual videos in the Pentagon’s secret vaults: raw footage from encounters between fighter jets and “anomalous aerial vehicles” — military jargon for UFOs.
Head of Pentagon’s secret ‘UFO’ office sought to make evidence public