I had a question regarding CNF (Chomsky normal form) in formal language theory.
I noticed that a lot of authors (including my own professor, and the Wikipedia page for CNF) frown upon or don't allow the start symbol to be on the right hand side of the production. However, I just can't wrap my head around why this is so.
In these cases, they put as the first step of converting a general context-free grammar into CNF to add the production
$$S_0 \rightarrow S$$
but I noticed that in some cases this actually leads to having one more unnecessary production, and in other cases it's impossible to completely remove the start symbol completely from the right hand side of productions (excluding the $S_0 \rightarrow S$ production). Also, the textbook that I'm using (Formal Languages and Automata, 6e - Peter Linz) allows the start symbol to be on the right side of a production.
What's the reason that this production is frowned upon?