"The bottom was a black piece of plastic with a small, 4-5 mm, recessed nut. The nut had a small amount of screw protruding."
This nut is what you need to remove - it holds the whole assembly together. Remove the gunk around it with a toothpick or such, then remove it with sturdy needle-nose pliers. The bottom will slip off, then the aluminum tube, exposing the mechanism. There are alignment tabs for reassembly, then screw the nut back on. The slidewires can be polished carefully with an eraser, then wiped w. q-tip and alcohol.
'91 MB 190E 2.3
'08 RAV4 Ltd 3.5
'83 Lazy Daze m'home 5.7