A Moment Generating Function (MGF) of a random variable $X$ is represented by an exponential
$$M_X(t) = E[e^{tX}]$$
Why isn't an MGF represented by binomial expansion?
For instance, $$M_X(t) = E\left[\frac{tX}{1-t}\right] = 1 + (tx)^2 + (tx)^3+ ... ... ... ...(1)$$
Hence, the derivatives are going to give the expected results.
So, what is wrong with binomial expansion?
$M_X(t)=E[e^{tX}]$ has the property $\left. \frac{d^n}{dt^n} M_X(t) \right |_{t=0}=E[X^n]$ (when it makes sense). $N_X(t)=E \left [ \frac{1}{1-tX} \right ]$ has the property that $\frac{1}{n!} \left. \frac{d^n}{dt^n} N_X(t) \right |_{t=0}=E[X^n]$ (when it makes sense).
Either one would work for computing moments of bounded random variables. $N_X$ is not finite in a neighborhood of $t=0$ whenever $X$ is an unbounded random variable, whereas $M_X$ is finite in a neighborhood of $t=0$ whenever the tails of $X$ decay exponentially fast. $M_X$ also shows up explicitly in certain situations, for example in Cramer's theorem. $M_X$ also has at least one other very nice algebraic property that $N_X$ lacks: $M_{X+Y}=M_X M_Y$ when $X,Y$ are independent.