I don't think that's really what means are. That intuition might fit the median better. One reason means are nice is that they have really nice properties, e.g. they're linear under addition of random variables. That makes them particularly easy to compute with and/or prove theorems about. Another reason means are nice is related to betting and the interpretation of a mean as an expected value; the theorem justifying this interpretation is the law of large numbers.
Nevertheless in many situations the mean of a random variable is a very bad description of it... (read more)