I hope to come out with an update soon that fixes this, sorry for the delay. Until then, here is the manual solution.
1. Open a command prompt. Press your windows-key and R at the same time. Then, type "cmd" and hit enter.
2. Get the command prompt to animation buddy's DATA FOLDER location (where yt-dlp.exe is). You would replace the path in the command to wherever animation buddy is, then add \AnimationBuddyData:
cd /D "C:\this\will\be\the\path\to\animation\buddy\is\AnimationBuddyData"
3. Run youtube downloader on your video. Type "yt-dlp.exe -S ext [YOUTUBE URL GOES HERE]" and it should give you an mp4 that you can drag into Animation Buddy.
I'm working on a fix right now. I suppose the YouTube and the downloader ecosystem changed; I was able to recreate problem (It seemed to worked for a moment, downloading, but then randomly failed). I believe If I auto-detect the default browser and pass it to yt-dlp.exe to use the same login cookies, it will work. (Otherwise users wouldn't be able to download age restricted videos).
Edit: There's currently a problem with the youtube downloader that would only allow this on firefox... https://github.com/yt-dlp/yt-dlp/issues/10927
So, I'll try to support videos with age-restriction for firefox, for now, but we'll see how well the youtube downloaer works. Perhaps it could be re-enabled for all browsers in future; their solution seems hacky. It will definitely work with regular videos in an upcoming update.