Engineer's perspective
Hey Ergrim, I know these things can be really infuriating, and without knowing everything about your system, I hope to help by offering a few things:
My old adage that has saved many a bacon is simply asking yourself this one question, "What did I change last?"
You may not like the answer and you may even be tempted not to answer it truthfully. After 39 years in engineering, honestly answering that question and performing the requisite follow-up has fixed many a "it's ***DEFINITELY*** not that" scenarios.
My most recent bug-a-boo was with the TrackIR interfering with a custom key setting with my Corsair keyboard.
So, for your joystick:
First, remove ALL unnecessary USB devices from the system - UNPLUGGED. I'd even go as far as leaving the mouse off and using keyboard shortcuts for windows menus. With all unnecessary devices out, take a look at the joystick operation from the device's management console.
Still not working right?
Next, try uninstalling the device from the Device Manager, even if you've done it before. Once removed, shut the system down and fire it up again. (e.g. don't just reboot.)
Bring the system back up without the controller plugged in. Go into the Logitech menu and uninstall. Now reboot the system once it's uninstalled. I'd also double check using System Internal's suite "AutoStarts" to check to make sure nothing remains in the auto-start for the joystick. Look for Logitech.
[start the AutoStart program in admin mode if it does and remove the offending entry(s).]
Now, clean slate, re-install per the manufacturer's instructions.
If this does not clear up the issue, I'd contact Logitech technical support before wiping Windows.
Note: I have seen more weird issues caused by mis-managing your USB hub connections than you might imagine. e.g. Always place streaming / high bandwidth devices on the first 3 ports on a 7-port USB hub. The last 4 are cascaded ports and share one I/O port from the first of the two 4-port chips. It's usually a bad idea to mix streaming devices with interrupt devices. (e.g. You should avoid placing streaming devices like web cams on the same physical PC port (via a HUB) with something like a keyboard or mouse.)
Keep one HUB for streaming I/O and another for interrupt I/O and you'll have fewer issues.
Hope this helps!