To do it through the Transformation constraint, set your constraint this way (image below). Essentially, in the constraint, choose your controller object as Target, click on Rotation for both Map From and Map To. Under Map To, as you want X to control the Y rotation, choose Y Source Axis > X, choose the degree amplitude, you can set to -10° and 10°, for ...
You have delta transforms:
To clear deltas, press Alt+R(G, S) → F9 (Adjust last operation) → Clear delta
If you want to keep current position, I could recommend press Ctrl+A → rotation, then Alt+R → F9 → Clear delta to rotate back.