That looks very annoying, the engine is not accepting your selection whatsoever. And considering you have reinstalled the plugin, the engine itself, cleared the cache, and tested different UE versions, you have covered most of the available paths.
I encountered something similar a few months, a dropdown menu not accepting the selection, or not even opening, and it had the most bizzare cause: a conflict with an overlay, in this case, Discord’s. So, I would suggest checking in your system for any active overlays (nVidia Experience, XBOX Game Bar, the already mentioned Discord, among others), and disable them. Then, test mapping the input again.
If the issue persists, I think the remaining option is a full uninstall of all Epic soft, not just the engine, but everything, plus clearing the registry. Here’s how to do it:
First, uninstall the engine, launcher, any asset packs, etc
Next, open the registry directory HKEY_CURRENT_USER\Software\Epic Games\Unreal Engine\Builds, and delete any build entries
Finally, delete the folders %userprofile%\AppData\Local\UnrealEngine and %userprofile%\AppData\Local\UnrealHeaderTool
After that, no elements related to UE should be left in your system
Now, reinstall UE launcher from scratch, and only add the latest version of the engine (in this case, 5.7)
Re-add the input plugin, then open a blank project in 5.7, and test input mapping again