I wouldn't say that the existence of the normal distribution is necessarily the reason that we place significance on the mean and variance. The mean is incredibly natural to define when you move to measure theoretic probability (simply the integral of a function, i.e. a random variable, with respect to some measure). When you take this point of view, random variables with particular moments existing are simply functions in L^p. Further, when you move on to proving the CLT the moments give you properties of the characteristic functions that allow you to prove the CLT. These are all deep connections.

There's another interpretation of the mean (and conditional expectation) as quantities minimizing squared error. It's not surprising that squared error and variance are so similar and that these are connected.