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JDownloader2 asking for donations to keep it "free of advertising" while showing an ad on the same screen
(feddit.org)
It also says "free of installer bundles" while (at least in past, don't know about now) their default windows installer was including ads (source)
P.S.: If you want to disable the in-app ads, here's how to: https://gist.github.com/yunooooo/87564a0293a1c7a53fb3e233a604c638
Does that one have something like Click'nLoad and a Browser add-on or app to allow for it when the Downloader is running on a remote machine? Can it run on my server and be accessed through a browser or said app?
Does it handle various hosters including debrid?
I'm not sure about Click'nLoad or Debrid, but aria2 does all the other things you just listed.
Try Brave.
What difference will switching to a new browser make?
I do not know the answer to some of your questions, but it can handle download links directly from the browser