Linear Regression: Why use global basis functions instead of local basis functions
I see a couple of parts to your question

Part 1


  Why would we use global basis functions instead of local basis
  functions to introduce non-linearity into our linear machine learning
  model?


Local (non-global) basis functions are used sometimes. For example, radial basis functions (RBF) are sometimes used as a basis in linear regression (see these lecture notes). 

However, often it matters that a model is interpretable and more complicated models are harder to interpret / draw certain conclusions from.  For example, imagine you're doing a controlled experiment, you may not care about "just" predicting some variable from (the variable you're manipulating + the variables you're controlling for).  But rather, how much of an effect your manipulation has on the output.

Part 2


  Why don't we discretize our feature space and optimize on a linear function space?


RBFs kind-of discretize feature space.  People don't usually literally discretized feature space because discrete optimization is much much more complicated than continuous optimization.