Newton or Einstein or Newton?

Recall this post.

A physics student of today will to pass exams have to say that Einstein's theory of gravitation is more precise than Newton's theory of gravitation, not less precise. 

The evidence supplied by the book is that the elliptic orbit of Mercury around the Sun shows a slow turn of main axis (precession) of 5600 seconds of arc per century, while Newton predicts 5557 taking in the influence of all other planets and celestial objects, and the missing 43 is predicted by Einstein as a small correction to Newton.  

In Einstein's theory the speed of gravity is equal to the speed of light in vacuum, while in Newton's theory the speed appears as being infinite. 

In a two-body problem of one small planet and a big Sun, the planet will according to Newton follow an elliptic orbit without precession, because the gravitational force on the planet acts in the direction of the Sun without delay from finite speed of gravity. With a finite speed of gravity there would be a delay which would make the planet slowly orbit away from the Sun. Massive observations of elliptic planet orbits show perfect agreement with Newton's theory. Here Einstein's theory can only be less precise.  

Finite speed of gravity thus appears to disagree with observations, at least in a Newtonian setting. To save Einstein's theory it is necessary to show that it contains some form of non-Newtonian effect which exactly cancels the delay effect of a finite speed of gravity. It is doubtful that this is possible. 

Newton's theory appears to be perfect (except for Mercury). Einstein's theory thus appears as a less precise version of Newton's theory (except for Mercury), in contradiction to the above book answer: Newton is more precise than Einstein. 

Concerning the precession of Mercury it is natural to ask: 

• What is the accuracy of Newton's prediction of 5557? Is everything taken into account?
• What is the accuracy of Einstein's prediction of 43 as a correction to Newton? 
• In order to match the observation of 5600, it is necessary that both 5557 and 43 are correct. Possible?   

If Einstein is less precise than Newton, what does that say about modern physics vs classical physics?

What did Einstein mean by saying "Newton, forgive me?" What what so wrong that an excuse was needed? 

To correct Newton requires a full understanding of all aspects of Newton's gravitation, and then a full understanding of Einstein's gravitation as a correction. Is there any living physicist with this intellectual capacity?

The Nobel Prize in Physics 2017 was awarded "for decisive contributions to the LIGO detector and the observation of gravitational waves" in the form of an extremely weak signal from an extremely massive collision of two black holes at an extremely large distance 1.3 Billion light years away from the detector. This was viewed as extremely strong evidence that Einstein is correct.