This is less a call for support and more a general notice to anyone who happens to run into this problem.
The new popular 'royale' shooter from EA - Apex Legends - has some kind of anti-cheat system that interferes with DOpus. After playing AL, all DOpus system-wide hotkeys stop working until you sign out and back into windows.
This problem might crop up in other EA/Origin games as well, so I just thought I'd mention this.
Depending on what their anti-cheat is doing, you may also be able to reinstate the hotkeys by fully exiting Opus (via File / Exit Directory Opus) and then restarting it.
Some anticheat don't like DO's files. I don't have that problem with Apex (probably because I do not use DO hotkeys but with Fortnite it was giving me another type of error unless, as stated above by Leo, DO was completely shutdown (not only "closed" from view but fully "switched off").
I believe "my" solution was to tick a box
_Settings > Preferences > Launching Opus > Startup > Shutdown > Tick "Shutdown DO when the last Lister closes.
Might also have to disable a few things under:
Settings > Preferences > Launching Opus
All these settings, I believe, will keep a file (DLL?) actively "open/loaded" in system which anticheats wrongfully interprets as a "menace" or something.
If Opus has to be shut-down, it's probably a system-wide hotkey that the anti-cheat objects to. Maybe those are also used by some macro-based cheats and the anti-cheat software blanket objects to all of them (which is pretty stupid, but not something we can change on our side).
If the Launching Opus settings also seem to matter, my guess is it's the "From the Desktop" setting, if any of them. The others probably shouldn't make a difference as they don't unload anything; they just change what some things do once loaded.
Any issues caused by anti-cheat false positives might be worth reporting to the companies involved, as I doubt they are intentional and they'll affect other tools as well.
I disabled DO's autolaunch on startup and the ability to start it from everywhere BUT the desktop shurtcut icon (or start menu) because I didn't know exactly which option would load/unload files that would trigger the anticheat.
Thanks for the precision regarding "only the Launching from Desktop" should interfere. Hopefully OP can confirm if only this setting will disable the trigger (as I no longer have Fortnite installed AND also that fact that although very similar, maybe OP's and my issue are not exactly the same (but may be fix by the same "solution").