Please note that I've read this question and it did not address mine.

I've been presented with the following argument regarding Taylor series:

We have a function $f(x)$, now assume that there exists a power series that's equal to it:

$$f(x)=a_0 + a_1 x + a_2 x^2 +\dots$$

One can quickly show, using differentiation, that

$$f(x) =f(0) + f'(0) x + \dfrac{f''(0)}{2! }x^2 +\dots$$

It seems that this argument implies two things:

For the Taylor series around a point to exist, it has to be continuously differentiable (infinitely many times) and defined at that point.

If the Taylor series for a function exists, then this implies it's equal to it, or that it converges to the original function at every point $x$.

Now I know very well that point 2 is false (not every smooth function is analytic).

But point 2 seems to be implied from the argument I presented above which assumes that if a power series exists such that it's equal to the function or in other words, converges to the function for all $x$, then it will be given by the Taylor series. So what's wrong regarding this argument above?