It's better to let your site cannibalize another site of yours

I have a pretty established education site, but I'd really like to make various changes.  The problem is that I have hard time coming up with a good way to transform it; I don't want to rebuild a new site and replace it.  So, I'm thinking about building a brand new education site from scratch targeting the same audience.  I'd like to make it better than the existing in time; in other words, I am aiming to let my new site cannibalize the old one.

Isn't it better to let your site cannibalize another site of yours than letting other people's sites to do that?  I think that the answer is 'Yes', and this is why I'm thinking about doing this.  What's your take?