Why is trace of energy-momentum tensor not a good candidate for the source of gravity?

The reason why I think trace of energy-momentum tensor can't be the source is that, in a static weak field $$p$$<<$$\rho$$

$$T^{\mu}_{\mu} \approx T^{0}_{0} = -\rho$$

and if in one system I have only $$T^0_0$$ component, this won't be true for all systems. Therefore, trace of energy-momentum tensor is not a good candidate.

According to what it was written in the following paper, it has something to do with coupling of gravity to electromagnetic field but I don't see how

http://www.cpt.univ-mrs.fr/~cosmo/SW_2013/PPT/Novello.pdf

"The source of scalar gravity is the trace of the energy-momentum tensor: this originates the main handicap of all previous scalar gravity: gravity does not couple to the electromagnetic field."