But the metric tensor of spacetime, in General Relativity, which is our best current classical theory of gravity, only explains gravity. Gravity, by itself, uses up all of the degrees of freedom in the metric tensor. There aren't any left for electromagnetism or anything else.

To get classical electromagnetism, you need to add another, different tensor to your model--a stress-energy tensor with the appropriate properties to describe an electromagnetic field. Of course doing this in standard GR is straightforward and is discussed in GR textbooks; but it does not involve somehow extracting electromagnetism from the metric tensor. It involves describing electromagnetism with the stress-energy tensor, i.e., with different degrees of freedom from the ones that describe gravity.(And if you want to describe the sources of the field, you need to add even more degrees of freedom to the stress-energy tensor to describe charged matter.)

The paper does not, as far as I can see, address this issue; the authors don't even appear to be aware that it is an issue. Which makes me extremely skeptical of the paper's entire approach.