You might test by installing only that JDownloader extension - installed in a separate Brave Browser user profile.
Create a new user Profile for Testing:
Create and use AN ADDITIONAL Brave Browser user Profile (Menu --> More tools --> Add profile) - only ← meaning: Do not have any other Brave Browser user Profile tabs/windows open.
When creating this new Brave Browser user Profile for test purposes
Do not import anything.
Do not add any extensions - other than the test extension.
Do not allow/enable any Brave Browser “bells and whistles” such as Brave Rewards, Brave Talk, etc.
IF this test Brave Browser user Profile does not present your problem, then possibly, the profile that you have been using, has a problem . . . and that might be caused by one or more extensions, or too many tabs ← complex/conundrum.
Detailed info about testing - IF, in regard to memory and performance issues:
I had the same problem. It was working in Chrome and Brave but stopped working in Brave a few days ago.
I had to set “Allow” for “Apps on device” in the site settings and now it is working again, even with Shields up and “Local network” set to “Ask (default)”.
Then it takes ~10 seconds an then switches to “green” and appears in Jdownloader.
After that a reload of the site comes to “green” at once with Click’n Load.
Wie der Zufall so will bin ich auch aus Deutschland. Nutze Brave normalerweise in Englisch aber hab es mal auf deutsch umgestellt für die Screens. Letztlich ähnlich wie in der Anleitung auf der JDownloader Seite, nur die Einstellung Apps auf dem Gerät, statt Lokales Netzwerk wählen:
For everyone else, I just activated “Apps on device” instead of “Local Network” as suggested in the Knowledge Base article on the JDownloader page. That did it for me. I don’t need to set “Local Network” or deactivate Shield.