
When you study the book: "Newton's Principia For the Common Reader" a large part is related to solve the differential equations i.e to find analytical solutions. For a system of two bodies this is an ellipse. The manybody problem is discussed in paragraph 62 at page 215. The threebody problem: the foundations of Newton's lunar theory is discussed in Chapter 14 at the pages 235267.
IMO what Newton's trys to find are analytical solutions of different n body situations. In general this is impossible. Only numerical solutions. In that sense it is quite easy to simulate all the planets around the Sun using Newton's Law. The problem is the predictions do not match observations.
In order to evaluate SR and GR you should try to do the same i.e. performing a simulation of all the planets around the sun, strictly using SR and GR and no approximations. This implies that you should also calculate the masses of all the planets. The reality is that this is that such a simulation is extremely difficult.
Newton is very much aware from the concept of time. Chapter 10a "The proportionality of mass and weight and the experiments on the pendulums" demonstrate the importance of time. The issue of what happens when clocks are moved is not discussed and is also of no direct importance for the study of the solar system.
What Newton does not discuss is that gravitational forces do not act instantaneous.
In Chapter 10 "On revolving orders" specific as part of Example 2 at page 196 he discusses different "forces of law".
(*) IMO when you want to understand the laws of nature you should start from the assumption that at any moment t any object considered has an instantaneous position x,y,z. Based on these instantaneous positions at many different moments t we can develop the necessary concepts and tools (laws) which are the basics to describe these positions. These instantaneous positions do not match with what we observe. That means certain mathematical transformations are necessary to transform what we observe into these instantaneous positions and to transform the predicted positions in what we should observe in the future. To perform these transformations the physical conditions of light (photons) should be considered.
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