I downloaded an age-gated video yesterday and it worked. You need to use the --cookies-from-browser option or export them to a text file. And I had issues with yt-dlp finding the correct path, so I needed to manually provide it with the correct directory for my Chromium or LibreWolf. My command looks like yt-dlp --cookies-from-browser firefox:~/.librewolf/ ...
It should be able to use Edge, at least according to the documentation you should be able to pass "edge" as the browser name without and additional info and it'll try to use them, unless your cookies are encrypted or you use profiles in your browser and it's not the default one.
But there are several bugreports for similar situations:
Maybe you want to install Firefox or something that works. Or if it's just one video, drop the link here and hope someone else downloads it for you.