Can we refute the Special Theory of Relativity using Newtonian view?

Someone may ask: Newtonian view and relativistic view are fundamentally different, how can we use one view to disprove the other?

My answer may not be satisfactory to you, but here is my reasoning anyway.

In mathematics, we first have addition and subtraction, then we get multiplication and division. As the latter two operations are the shortcuts for the former two operations, we expect full compatibilty among them. If any shortcut would lead us to the wrong destination, why do we use it?

Also in mathematics, there are both trigonometric functions and hyperbolic functions. They have similar properties, and have very similar names. For sin(x) there is sinh(x), and for cos(x) there is cosh(x), and etc. We can accept the fact that they are different, because the latter just borrowed the names of the former.

Let's come back to our topic here. Before the invention of the Special Theory of Relativity, Newtonian view is the only one available, right? Where did Einstein start from? He certainly did not start from a relativistic view. If it were already there, why would you take the trouble to create it again? The only possible start, as I understand, is the classic Newtonian View. The proof is in the first step of his paper. The c+v and c-v there all come from the Galilean transformation, not from the relativistic transformation.

It does not matter where you start from, but you have to make the theory self-consistent.