Can we represent a population by its mean? Can we represent a population by its mean?
Is the answer simply "yes, unless there are outliers or the data is skewed"?

Easiest Example
It may help to look at a visual example, which is always good practice anyway when trying to understand where your data is centrally distributed. The easiest example is simulating a normal distribution. Here I have done so, including a mean:
x <- rnorm(n=1000,
           mean=20,
           sd=10)

The mean should be obvious and plotting it makes it easy to see why.
plot(density(x), main = "Density of X Variable")


From here I will plot the mean in red and the median in blue. Here you can see they are exactly the same, which make sense because the data is pretty easily seen to be mostly distributed in the center of the graph.

When Means Become Less Helpful
Now the most obvious example of when means become less helpful for understanding your data is when the data is heavily skewed. Here I have plotted a chi square distribution with mean and medians.
z <- rnorm(n=1000, mean=20, sd=10)^4
plot(density(z),main="Density of Z Variable")
mean(z)
median(z)
abline(v = mean(z),                    
       col = "red",
       lwd = 3)
abline(v = median(z),                    
       col = "blue",
       lwd = 3)


You can see that the median in blue is a much better approximation of where the center of the distribution is, as the mean is being dragged away by the data points on the right.
So to answer your question, the mean in useful when it's practical purposes is useful. If the population does not fit the mean well, then forgive the pun, the mean may not mean so much.
