>Is it correct to say that the non-linear activation function's main purpose is to allow the neural network's decision boundary to be non-linear?

Yes.

Neural networks compose several functions in layers: the output of a previous layer is the input to the next layer. Linear functions are closed under composition, so the result of stacking several linear functions together is a linear function:

$$
\begin{align}
\hat{y} &= W_2(W_1x + b_1)+b_2 \\
&=  \underbrace{W_2W_1}_W x+\underbrace{W_2b_1+b_2}_b \\
&= Wx+b
\end{align}
$$

Any model which minimizes a loss $L(y,\hat{y})$ over parameters $W_1,W_2,b_1,b_2$ is equivalent to a model which minimizes the same loss over parameters $W,b$. In the special case that the loss is the square error loss, this is *exactly the same* as an OLS model. 

On the other hand, using a nonlinear function $f$ makes the map from the input to the output nonlinear:$$
\hat{y} = f(W_2 f(W_1x + b_1)+b_2) \\
$$
 When $f$ is some well-chosen pointwise nonlinear function, such as $\tanh$ or ReLU, this cannot be rewritten as a single linear operation on $x.$

The importance of nonlinearity to neural networks isn't unique to classification problems. If you have some sort of regression problem (such as an output that can take on any real number), then using nonlinear activation functions is necessary to model a nonlinear relationship between the inputs and outputs.

In both classification and regression settings, the purpose of an activation function is akin to basis expansion or regression splines, with the added flexibility that the neural network optimizes the weights to create the basis & reduce the loss. 

  [1]: http://www.argmin.net/2017/12/05/kitchen-sinks/