I am asking this on behalf of the Elementary School in my district. (I work in the library of our district's high school). The Elementary School participates every year in a program called PARP. (Parents As Reading Partners). The teachers and principal always make some sort of video to kick this off this event since pandemic times.
This year the entire school is reading the SAME book: The World According to Humphrey, by Betty G. Birny. (It's a story about a Hamster and how he deals with life issues). My district's teachers want to "borrow" liberally from this Animoto video: https://animoto.com/play/ICom40fpoTdMzDov931aDQ
This video contains four components: 1. Another School (We'll call it School X, an independent school in California essentially doing the same thing), 2. an interview with the author segment, Betty G. Birny, 3. an interview with a store clerk from PetCo and 4. a video of a hamster performing "cute antics" with a voice-over dubbed in called April's Animals. (This individual posts varied animal videos on YouTube)
What my teachers want to do is create their OWN video of teachers and the principal endorsing this book, interspersed with the hamster video from April's Animals. I did observe at the end of the Animoto video, there were credits provided. My school would not use the PetCo interview or the Author Interview or the School X video as those segments are directly related to that specific school. They want to do the same idea and only use the video provided by April's Animals. I didn't know if this would be problematic because we are a public school, this would not be posted on YouTube. It would be shown over our school network to our K-2 classrooms one time only.
The short answer to this question is: IF the video is only going to include the YouTube animal clips, and IF it is only going to be used in the school for instructional purposes, the proposed use is fine, since copyright section 110(1) allows schools to play videos in class if the topic is related to a class, and YouTube doesn't limit use of its service to "personal" uses.
Now, I say "mostly" fine because, technically, the combination of the YouTube content into another video compilation could be considered the creation of a "derivative work" (like a sequel or a mash-up), instead of just "performing" (playing) the video as allowed by law. But if the copy truly isn't leaving school grounds, and the "performance" is to promote a reading program in the classroom, and the footage really is just being swapping in and out with interviews with school staff, it would be a stretch for anyone to claim infringement.
With respect to the other issue that I detect in the question--would "School X" have a claim against the school for pinching its idea? I don't think so. The project you describe is sufficiently different from theirs; after all, they got their author for their endeavor, and your school is focusing on local talent. You can't copyright an idea...just its expression.
When it comes to a school generating original educational content inspired by others, for use only within that school, the key is to model the type of respect for others that educators want to instill in their students, while taking full advantage of the protections educators have under the law.
In this case, "Respect" means not using pirated copies when a school plays instructional movies, and not using more content than the school is entitled to when the instruction is online. "Protections," among other things, means that for in-person instruction, videos can be played, and for online instruction, parts of videos can be played, so long as the performance is from legitimate copies.
[NOTE: For schools that want to up their game and start producing original content they will share with the world: this answer is not for you. If any school out there is thinking of becoming an author/producer/provider of educational materials, don't rely on this answer, and develop a business plan that includes how to respect and protect IP.]
And finally, I have to say: thank you for this question. First, it got me onto Animoto, which I am totally going to check out. And second: I love PARP. Some of my fondest circa-1980 memories are of filling out my PARP form with my folks, after some time reading together on the couch, so this question made me smile. It's good to see the program is going strong, and the hamsters of the world are showing us how to cope with the ups and downs of life.
 110(1) allows "Performance...of a work by instructors ...in the course of face-to-face teaching activities of a nonprofit educational institution, in a classroom or similar place devoted to instruction, unless, in the case of a motion picture...the performance... is given by means of a copy that was not lawfully made...."
 Of course, you can patent certain ideas, so please don't think I'm touting intellectual property anarchy.
 This aspect of Copyright Section 110 is different than the issue of streaming services being limited for personal use, and thus not always the best place for educators to get their in-class movies.
 I clearly don't get out much.
 My parents still have the same couch, which they got in 1964. They are the greenest people I know.