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. If you compose linear functions, these functions are all linear. So the result of stacking several linear functions together is a linear function. Using a nonlinear function makes the map from the input to the output nonlinear.
For example, a ReLU function's output is either 0 or positive. If the unit is 0, it is effectively "off," so the inputs to the unit are not propagated forward from that function. If the unit is on, the input data is reflected in subsequent layers through that unit. ReLU itself is not linear, and neither is the composition of several layers of several ReLU functions. So the mapping from inputs to classification outcomes is not linear either.