Is Newtonian Physics a Sufficient Explanation for Celestial Mechanics?

Everyone has probably heard a physics teacher tell you that Earth moves around the Sun like a stone rotating on a string. Stone's inertia = Earth's inertia. String counteracting it = Gravitation counteracting it.

Youtube provides a video for those who haven't:

Earth revolves around the Sun like a stone on a string | Gravitation | Physics on KClassScienceChannel :

https://www.youtube.com/watch?v=QKvCmUpCVIA

Is this a good illustration?

If inertia = inertia is a good illustration, string = gravitation is perhaps not the best we can do. How about electromagnetic attractiong = gravitation?

This video provides an example:

[ISS] Don Petit, Science Off The Sphere - Water Droplets Orbiting Charged Knitting Needle on SpaceVids.tv:

https://www.youtube.com/watch?v=UyRv8bNDvq4

Note that the balance between inertia and attraction is more delicate than that between inertia and string. It breaks in this example after 10 - 20 orbits.

So, how can we be sure the balance is better in the cosmological case? That it works without either divine or angelic interference?