GetExeDirectory only works on Windows, so you'd need to make some sort of alternative if you want to use it. A change to make Dolphin use that by default instead of what it does now is unlikely to be accepted, but what you could try to do is to implement Dolphin's portable mode. On Windows, Dolphin will run in a self-contained folder (instead of storing configuration in Documents) if a file with the name portable.txt is present in the same folder as the executable. This is not implemented on other OSes at the moment, but pull requests are welcome. https://github.com/dolphin-emu/dolphin/blob/696d6eae099feba23c280017c5184785fc1e3076/Source/Core/UICommon/UICommon.cpp#L87
Being a Windows user, I'm not sure how the sys directory should be treated in portable mode on non-Windows OSes. Should people be able to put the config in the same folder as the executable without doing so with the sys folder? I suggest asking some other developers, preferably on IRC since not all of them use the forums. It's also fine to just make the PR and then see what they have to say about it, but you might have to rewrite some of the code then.
Being a Windows user, I'm not sure how the sys directory should be treated in portable mode on non-Windows OSes. Should people be able to put the config in the same folder as the executable without doing so with the sys folder? I suggest asking some other developers, preferably on IRC since not all of them use the forums. It's also fine to just make the PR and then see what they have to say about it, but you might have to rewrite some of the code then.