I submit that while most of your points are perfectly valid, in terms of 'why this and not that', our heavy reliance on prepositions in English make 'into' generally mean something quite different from 'in to'.

Yes, into is not the same thing as in to, but the to in to be or not to be is also quite different from the to in I gave the book to Jill and they're spelled the same.

Hope you stick around longer and post now that your crossword dilemma has been addressed.

Ceci n'est pas un seing.