Single Layer Perceptron vs Multi Layer Perceptron
Your premise is wrong.  A single-layer neural network (perceptron) can use a (nonlinear) activation function.  Nothing prevents you from doing that.

It's common that the last layer of a neural network doesn't use any activation function, and instead is input into a softmax layer.  If there's only one layer, that means that no activation function is used.  But this is just a standard convention.  There's no ironclad law that says you have to do it this way.  You could have an activation function after a single-layer neural network, if you wanted.