I've changed the key binding for the edit page, such as by hitting backspace (the default) or by hitting x then the selected clips are deleted. I made sure that DaVinci Resolve is using my custom saved profile and yet it is still not working.
EDIT: I was able to assign and de-assign 'c' to be the same as backspace. That's weird. When I added 'c', it showed up on the right of the backspace in your screenshot, but when I closed and reopened, it have moved to the left as in your screenshot. I'm using 16.2.3 on a Mac. Sorry I don't have any tips. Maybe your preferences are corrupted somehow?