Is it okay to include a simplified and easy version of a result already proved?
Yes.

New proofs are always good and of scientific value, and may be of significant use and interest in their own right.  In some cases they are even worthy of a paper of their own.  As an example, just recently on MathOverflow I saw Zhi-Wei sun mention that a result of his currently has 6 proofs from 6 different papers and sets of authors (including his own and himself).