The provided example is just to get across the intuition for how IBS/IBE works.

For production, the Boneh-Franklin construction based on Weil pairings is a much better choice: https://crypto.stanford.edu/~dabo/papers/bfibe.pdf

This paper is also very interesting: https://www.researchgate.net/publication/334001157_Extending_the_BLS_Scheme_to_Identity_Based_Signatures