I have another speedo with a non-functional odometer and the needle is rock steady so the problem is the speedo.
The face does not effect it but the pin on the face holds the need at 0mph. if you remove the pin and the needle is under spring tension, it my rotate freely and if it does this for 360 degrees, when you put the face back and set the needle back in front of the pin and fail to rotate it back, the tension is less. I am going to rotate the needle one revolution and see what I get.
I have surgical tools, in particular a small "Kelly clamp". if you remove the screws on the face, you can pivot the face , use the Kelly and remove the screws holding the speed together. No need to remove the needle.