Removing the reflection in post us going to be rough… Lots of careful dodge and burn and cloning is about as good as you’re going to do.
For the capture, @darix is right, polarizer on the flash and on the camera will get rid of the glare. You’ll need to do a bit of fiddling with flash power and exposure to figure out how much light the polarizers eat up.
This is common practice in commercial photography for thingsine watches and jewelry
As a former commercial industrial photographer I would also agree that the solution is based on using a polarizing filter on the camera. I personally never polarize my light source as it was not practical for me, but you could try that. Instead I would place the light source at an angle such as 45 degrees. If the light is coming from the direction of the camera then no polarizer will remove the reflection. This is because polarizers work best when the light is ninety degrees to the direction of the camera (side lit). Sidelight is inappropriate for most subjects, hence my preference for 45 degrees.
With subjects like in the OP I would bounce the flash off a white card and would not expect much refection even if the flash was mounted on the camera. With a diffuser stuck on the front of the flash I would still expect reflection. I am not a huge fan of the diffusers that are supplied for the flash units.
In my stilllife studio days in very rare cases we would use dulling spray to make hard reflections smoother. We never used a polarizer, anywhere.
It was all about controlling the light and using it to get the effect you want.
We had a huge collection of every kind of diffusing and reflecting materials to shine through or bounce off the lights. The lights rarely had anything else than the standard reflector on them.
For stills … fix the object and the camera for a good shape. Only then start working with the light.
If you use a flash/speedlight without modeling light use a torch/flashlight to test out how your object reacts to different positions. Fix the light. And then start modeling it.
Btw, a lot of controlling the light is actually about controlling the shadows or to be even more precise: the transition from lighter to darker areas.
Gradients are the magic stuff in stilllife photography.
Are you using it on the camera? If yes, try pointing it upwards and sideways - or even backwards - and bounce the light off some white paper. That will increase the relative size of the lightsource compared to the object significantly and should improve your lighting a lot.