I have a potential solution (at least, for wired controllers and keyboards and without using CUE software). I just discovered this solution, however, so it's only preliminary. I will keep this post updated.
So I just got a wired K63 keyboard, and noticed that my DS4 controller stopped working. I didn't install any of the CUE software or anything like that, because I primarily wanted my keyboard just for the feel of the keyboard switches (and everything worked fine out of the box).
I went to devices and printers from the Windows control panel, and saw that a "Wireless Controller" was listed, although it had a yellow (!) icon indicating something was wrong with it.
I right-clicked the "Wireless Controller" and went into the properties, and then into the "Hardware" tab. One of the drivers was disabled. I re-enabled it and my DS4 (wired) began working again.
In order to re-enable your disabled gamepad drivers (the statuses are listed in the Device Function Summary):
1. Right-click the driver and click on "Change Settings" on the bottom-left corner of the first window that opens.
2. Go to the "Driver Tab."
3. Click the option that says "Enable Device."
Now, even though I didn't disable or uninstall any of the corsair drivers, my controller works.
EDIT: Even after turning my computer and starting it back up, my gamepad still works.
EDIT2: Seems the driver can still occasionally become disabled after restarting my computer, but it's still very easy for me to just re-enable the driver.