(e.g. there are no rumble effects with this gamepad in "Duck Tales: Remastered", although the game options offer this as a controller setting.)
Try disabling Gamepad Configuration Support in the Steam client options. Let us know if that doesn't work.
Ok, disabling "generic gamepad configuration support" makes the rumble effects work.
(Great! That means that I can now have rumble effects with gamepads that are supported in Linux, even if the same device has no rumble device driver in Linux... Yay!)

Cool, thanks for confirming. This is due to a Steam client limitation, see https://github.com/ValveSoftware/steam-for-linux/issues/4338 . I've updated the Proton FAQ on the Wiki.
This is really cool, as I own a gamepad that rumbles on Linux, but does not on Windows. :-D
@aeikum It would be really cool to have this work out of the box without changing any setting.
Agreed, but it's a Steam client issue, and I don't work for Valve. Head to their bug tracker :)
My very cheap
0079:0006 DragonRise Inc. PC TWIN SHOCKgamepad comes with force feedback effect vibration motors. I can confirm that these work under Linux withfftest. What is the proper setting for Steam with Proton to enjoy rumble effects?