I don't know whether to be shocked or impressed by the prices that those watch winders sell for, given how cheaply they are made.
The first things that I would check are the switches and buttons, as those are things that can easily fail due to contact corrosion and are reasonably easy to replace. Behind the plastic knobs or buttons will be some circuit mounted switches which can be unsoldered and replaced.
I recently did that for a DAB radio that was kept near the sea, and the buttons went intermittent after 10 years or so.