The drawback in this reasoning is that with Newtonian mechanics alone you cannot know whether your measurements are in the domain for which it is accurate or no

The drawback in this reasoning is that with Newtonian mechanics alone you cannot know whether your measurements are in the domain for which it is accurate or not. For that you need relativity and quantum mechanics. Newtonian mechanics alone is seriously wrong, but if you know (because you understand relartivity and quantum mechanics) that you are in a domain for which it is sufficiently accurate, then you can use it to produce accurate results.