Gravitation: Newton or Einstein?

Modern physics is based on an assumption that Einstein's Theory of Gravitation EG in the form of his General Theory of Relativity gives a more precise description of the true physics of gravitation than Newton's Theory of Gravitation NG. 

Is this assumption justified? What is the evidence that EG is more precise than NG? 

NG based on Newton's 2nd Law and Newton's Law of Gravitation (inverse square law) combines maximal generality with maximal formal theoretical simplicity allowing computational simulation of gravitational interaction of billions of stars/planets over billions of years. 

EG on the other hand comes with maximal theoretical complexity making computational simulation impossible for gravitational interaction already for 3 stars/planets. 

Is it then possible to verify that EG is more precise than NG? If EG is uncomputable? 

The prime evidence that EG is more precise than NG, is a back-of-an-envelope computation by Einstein in 1915 concerning the precession of Mercury showing a correction to a computation by hand using NG made in 1888 by the astronomer Simon Newcomb supposedly taking into account all the effects from the other planets, with the result of 5557 seconds of arc per century (one second of arc=1/3600 degrees). The observed precession was 5600 and Einstein's back-of-an-envelop computation came up with exactly the missing 43 arc seconds per century, which still serves as main evidence that EG is more precise than NG.

How convincing is this? Questions line up:

1. How precise is the computation by hand by Newcomb, supposed to account for all effects in the Solar system with its planets, moons and asteroids swirling around the Sun? Has the number 5557 been confirmed by best possible computation today? If so to what result? Exactly the same as Newcomb?

2. Einstein knew that 43 arc seconds were missing and so could target his correction to fit exactly. Convincing?

3. It is impossible to directly compute the precession by EG. So Einstein starts with the 5557 given by Newcomb using NG for the whole Solar system as a complex many-body system. Einstein then isolates to the two-body problem of Mercury + Sun with EG offering a correction to NG which precisely matches  the missing 43. Magic?

The weakness of Einstein's argument that EG is more precise than NG, is that direct computation with EG to this effect is impossible. It is only possible to start from a NG computation of a complex many-body problem and then isolate to a two-body problem for which EG appears as NG with an extra contribution to potential energy and use this as a correction to the many-body problem. 

It is obvious that this procedure has some weak points. Questions pose themselves:

• Is its worthwhile to spend years of study to come to at least some understanding of EG, when EG is severely uncomputable?
• Is the evidence that EG is more precise than NG convincing?
• Is it reasonable to view EG as a more precise version of NG, when only NG is computable?
• Is it reasonable to use EG as foundation of modern physics when EG is uncomputable?
• Is it reasonable to use EG only as a form of decoration, which serves no practical use?
• Is it reasonable to give up the basic concepts of space and time of Newtonian mechanics, which have served and continue to serve science and society so well?
• Is it a good idea to insist on EG when EG is incompatible with quantum mechanics? 
• Why was EG initially met with very strong skepticism?
• Why was EG accepted only after Einstein's death (and of all his original skeptics)?