Is feature scaling at all needed for a feature set with a single feature?
I understand that feature scaling is required to bring features in
different magnitudes on a common scale so the model is not biased
towards features with higher magnitudes. But if there is only a single
feature in a feature set. For example, I have a quantity (events over
time) on a time series and I'm forecasting for the future. In this
case, do I need feature scaling (normalization or standardization?) or
will it be unnecessary?

Usually we would try to do feature scalling although the idea is to remove the dependence of absolute magnitiude in affecting  convergence. it will help if the distribution is made gaussian with zero mean and unit variance. Since the gradient will be lower the stochastic descent will be a lot smoother.