That's the thing. They can be an anti-machina faction without the name 'New Yevon' Why would you want to be associate your faction with a name that should be universally reviled? And only a short time after Yevon have been defaced. More importantly, how the hell did they succeeded?
You just can't disassociate religion (can be anything actually) to people especially if it's deeply embedded in their culture/way of life. It also applies to real life like in politics where a slight change in name or usage of a popular term is all you need to prove that in essence, you are still the same party even with a change in perspective.