and a software keylogger requires direct access to the keyboard driver, which requires administrator privileges on any sane operating system
No it doesn't.
To access the actual keyboard, maybe. But you only want the key events for a key logger; and that certainly doesn't need admin access. If it did then how would what you were typing ever make it to the application?
Key events can only be received by whichever application has keyboard focus. They cannot be accessed by other programs without admin privileges.
I'm sorry, but that just is not true. In Linux, for example, the window manager, a separate app, can pick up key presses like "Alt-F4" before the app to close the window -- any it doesn't want it simply forwards. At the extreme end on UNIX you could just catch every byte sent to the X window control socket (which is owned by the user, not root). But you could also make one giant transparent window that receives everything and simply passes it through to whatever app underneath the keylogger wants.
Have you ever seen virtual keyboard apps? Or accessibility helpers? Both of those get at the keyboard without any difficulty.
Sending key events and receiving those sent to other programs are two different things. The latter is not normally possible under any sane operating system. If you're asking about the possibility that such programs might log their own input, then yes, this is possible, but entering passwords directly into another application is kinda dumb.
That's why I mentioned two classes of program. Virtual keyboard apps can send; and accessibility helpers can capture keys (slow modifiers is a common one). Combine those facilities in one application.