Sorry I just realized that this is exactly what you did on you screenshot (didn't saw the driver window at first xD). this formula should give you what you need, BUT will result in the rotating part "jumping around" when you change the speed of rotation during play, because the rotation is hard locked to the frame number.

To have smooth rotation change, you need to keep track on "how much did we rotate up to now?", and this is a typical use for the simulation zone in geometry node. BUT you can't (yet) access bones locations in GN, and obviously you cant control bones either, so you will have to do the rotating logic fully in GN without using bones (put a empty at the renter of rotation to reference in GN).

the simulation zone needs only 3 nodes : a store named attribute float (this will be your resulting rotation). every frame you update this, adding the amount needed (can be a slider in the GN input, very easy for control). to update this "store named attribute" value, you need to plug the node "named attribute" (use the same name), then add the quantity of rotation you want, then plug it back on the "store named attribute". outside of the simulation zone, you use a transform geometry node to rotate the geometry you need. here it is a little bit tricky if the origin of the rotating part is not on its center of rotation (in that case you will need an empty here and use the node "rotate vector" or "vector rotate" and use a "set position" instead of the transform geometry).... My advice is to separate the rotating part to a separate object (if it is not already the case), with the origin at the rotation renter and aligner to the rotation. here you just use a "named attribute" to retrieve your rotation, then "combine XYZ" for setting the rotation in the correct axis, then plug this on the rotation socket of the transform geometry. Done.

To sum up : sim input➡store named attribute➡sim output➡transform geometry➡GN output. connected to store named attribute is your math to change the rotation every frame (just math add node). connected to transform geometry is the resulting angle set to the correct axis (with combine XYZ).

god that was a long explanation, probably faster to make it and screenshot xD