"Simply modifying" Newtonian gravity to have it spread at finite speed does not work if the finite speed is the speed of light. It was attempted by Laplace in his Celestial Mechanics (1799), who found that the planets will promptly fly off their orbits and the Solar system will disintegrate in seconds, unless the propagation speed is 7×106
times greater than the the speed of light. This is because of the aberration of the direction of attractive force due to delay in transmission, see Resolving General relativity and Newtonian mechanics on a computer.

A more sophisticated modification follows from Mossotti's electromagnetic gravity hypothesis: electric attraction and repulsion do not balance each other exactly, and the difference is gravity. In 1864-72 Seegers, Scheibner and Tisserand experimented with applying the velocity and acceleration dependent correction to Newton's law imported from Weber's electrodynamics to the precession of the perihelion of Mercury. Around 1900 Lorentz, Einstein's precursor on special relativity, showed that under the Maxwell electrodynamics the Laplacian aberration problem is eliminated because the correction is of the order v2/c2
rather than v/c
that Laplace assumed, so the attraction between masses moving with constant relative velocity is always toward the instantaneous position of the other mass. It is the Lorentz invariance of the Maxwell electrodynamics that cancels the effects of transmission delay to the first order, as Poincare pointed out in 1905. See What 19th century developments contributed to the General theory of Relativity?

However, Lorentz's theory did not work either, and this time exactly because of the perihelion of Mercury. //

Einstein first mentions Mercury in a letter to Habicht in 1907:"At the moment I am working on a relativistic analysis of the law of gravitation by means of which I hope to explain the still unexplained secular changes in the perihelion of Mercury."