Speed of Gravity in a Static Gravitational Field?

To save General Relativity GR assuming that the speed of gravity is finite equal to the speed of light from collapse when confronted with observations apparently requiring a very much larger speed of gravity, it is commonly stated that in a static gravitational field there is no effect of time delay from finite speed of of propagation. And so common wisdom claims that there is no contradiction between GR and observations conforming to a speed of gravity much larger than the speed of light. 

It is this convincing? No problem in GR from finite speed of gravity? Let us see: A gravitational model with finite speed of propagation $c$ in a Newtonian approximation of GR takes the form

• $\frac{1}{c^2}\ddot\phi -\Delta\phi =\rho $,       (*)

where $\phi (x,t)$ is the gravitational potential corresponding to a mass distribution $\rho (x,t)$, where $x$ is a Euclidean space coordinate, $t$ a time coordinate and the dot signifies differentiation in time. 

Now a static gravitational potential is characterised by $\ddot\phi =\dot\phi =0$ and so the value of $c$ can be anything, in particular as large as desired even larger than the speed of light without changing anything. In other words it is meaningless to speak about speed of gravity in a static gravitational field. 

To state that in a static gravitational field in GR there is no effect of finite speed of gravity does not make sense. There is no speed at all. 

Further, gravitational fields are not static, not even between the Sun and Jupiter, and so this case lacks interest. 

Yet in GR the speed of gravity is viewed to be finite = speed of light c, which requires a theory of quantum gravity to explain finite speed. But no theory of quantum gravity has been found despite intense search for 100 years. Further, gravitational waves in GR are viewed to require merge of black holes to appear... 

The idea of a finite speed of gravity = speed of light is the main road block to a Theory of Everything ToE combining Newton, Maxwell and Schrödinger. What would happen if we simply remove the block by replacing Einstein by Newton? What would be missed? Nothing? And then?