Yes, gravity could be used to communicate faster than light if we could find a way to do so.
That was actually part of the story to Interstellar. How Cooper could communicate with Murph via the manipulation of gravity while he was in the middle of the black hole.
Excellent post. The concept of this sort of communication first came to my own awareness in reading Ursula K Leguin's SF writing. Her "Ecumenical" novels (Left Hand of Darkness, Word for World is Forest, The Wind's Twelve Quarters" et al) propose that there is a sub-ether in the fabric of space which could allow one to communicate across space over great distances instantaneously, but not to transport material objects faster-than light.
SF author Gene Wolfe also postulates that strange things can happen to spacetime when massive acceleration or massive energy are applied to small regions using very precise mirrors (taking the concept of laser amplification into the trans-dimensional realm) in his Urth of the New Sun novels (aka The Severian Series). His novels are also full of conservative themes, the latter series being very analogous to the life of Christ in some interpretations.
