Taylor series (or Maclaurin Series) are the only way to get values for some functions
This is not true in the example you give. There is at least one other way to get the value of erf, which is to do numerical integration of the integral you wrote down as a definition.
BTW, you don't need to say "Taylor series (or Maclaurin Series)," because a Maclaurin series is a Taylor series.
how do we know this convergence is to the actual number this function should give if we can't check by another means?
Well, if the only information you have about this function is its Taylor series, then you can't determine whether the Taylor series converges to the correct value (at a point inside its radius of convergence) -- because you have no other information about its correct value.
I'm sure there are functions we can define such that nobody on earth can prove any nontrivial facts about exact values of the function. For example, let $F_n$ be the nth Fibonacci number, and define the function
$$f(x)=\sum \frac{x^n}{F_n!}.$$
This function is analytic everywhere on the real line. I don't know, maybe someone can prove something about some exact value of this function other than the trivial fact that $f(0)=1$, but since I made this example up essentially at random, it seems unlikely.
But very few real-world examples are like this. In most cases, we have some reason why we're interested in this function, which implies other things about it. E.g., some books do define $e^x$ in terms of its Taylor series, but then they prove things like $(e^x)'=e^x$ and $e^{x+y}=e^xe^y$ based on that definition. This gives you a body of facts that can all be correlated with one another.
Especially for functions with huge radii of convergence, why should the students expect derivative information taken around a single number to give accurate values extremely far from that number?
In general, they should not expect this. Analytic functions are in some sense just a infinitesimal subset of the set of all functions. (WP gives the following more rigorous statement of this fact: "And in fact the set of functions with a convergent Taylor series is a meager set in the Fréchet space of smooth functions.") But many of the important functions we use a lot in math are ones that have nice properties, and the nice properties are the reason we study those functions to study. Once such nice property is if a function is analytic.
There are certainly techniques for proving whether certain Taylor series converge to certain values, but they may not be appropriate to teach in a second-semester freshman calc course. For example, if I'm remembering my long-ago complex analysis correctly, then the function $1/(x^2+1)$ is going to have a Taylor series about $x=0$ that converges to the correct value throughout its radius of convergence of 1, and this is because it's formed by the composition of functions that are analytic except at $x=\pm i$.
One can certainly say things about the error incurred by truncating a Taylor series, e.g., putting bounds on this error. But I don't think this has much to do with your question, since functions like $\exp(-1/x^2)$ would have small bounds on the truncation error, but the error relative to the desired value is large.