The image is semi-transparent even before you export it. You can use the Pointer dialog to check the alpha-channel.
Color-to-alpha will always make things as transparent as it can. The "contract" is that after you have used C2A, if you insert below the edited layer a layer filled with the color you removed, you get back the original image. However, there are many ways to achieve this. When you remove white from light gray, should the result be opaque light gray, somewhat transparent dark gray or very transparent black? Gimp picks the most transparent solution every time.
So to avoid Gimp altering areas that must remain opaque, just exclude them with a selection. Making a selection that doesn't include the center of the subject, but only its edges is easy:
- Wand-select the background
Select>Grow by one or two pixels so that the selection includes the edge pixels
- Use Color-to-alpha to remove the background