yt-dlp is generally the best open source tool for downloading/backing up videos from the internet. Despite the name, it works with thousands of video streaming sites, not just youtube. Completely free. I would never pay for a video downloading software.
It is command line based though. I'm sure there are forks with a UI, but I'm not familiar with them. There is a fairly detailed wiki for it, and a discord support channel. The defaults are fine for youtube back ups.
I would reccomend uploading whatever you back up to the Internet Archive. Then you aren't responsible for hosting, and it's available to the public.
A tangentially related but good example of this sort of thing is BluRays and community movie nights (like setting up a projector in a park).
Most of these movie nights are de facto illegal, as even though you own the BluRay, it is not licensed for public showings, just for personal use. Obviously no one gives enough of a shit to enforce this against small groups, especially if they aren't making money off it, but if a theater started offering showings of shit the owner just bought on BluRay or UHD disks, it wouldn't last too long.
Similar thing here. Just because you can access the content to view it yourself doesn't mean you have the rights to do more than that with it. As an individual, you're likely fine to break those rules. As a giant fucking corporation, it's time for you to pay up.