If it’s just videos you want, you can try using network inspector to see if you can catch the url of the file - assuming giving the url of the video’s webpage to youtube-dl along with a snapshot of your browsers logged in cookies doesn’t work. You might also see an m3u8 in the network inspector, which you can also give the url of to youtube-dl and it’ll download all the segments and merge them into a video file (you might also need auth cookies or headers unless it’s a temporary url which can work anywhere, just check the network request to see what’s sent). Some sites do separate m3u8 for video and audio or multiple ones for different video qualities, so you might need to change the quality to maximum for the browser to request the high quality stream url. You might also see a file requested that just lists the urls for m3u8s of each quality. If you see a vtt file then you can also grab that, convert to an srt, and remux with mkvtoolnix to embed it into the file as an optional subtitle.
This should all work as long as they don’t use drm / widevine type stuff and as long as they don’t have some supremely annoying security measures (like using authenticated urls that are one time use so by the time your browser shows it in the network inspector the url is expired or something). Otherwise for widevine you’ll need to do some kind of screen / HDMI capture type setup.
This might be too specific and may not suit your needs, but I used ArchiveWeb.Page to download entire webpages without any of them breaking. You can record webpages and save them as a .warc file. You can open the .warc file using the ReplayWeb.Page app. Both apps are open source and are meant to be website archivers but also works great for this purpose.
I tried using extensions that can save a webpage as HTML/PDF/whatever, but that often didn’t save everything in the webpage and the downloaded copy is pretty much fully broken and useless.
