Here is a list and comparison of companies that do this, in case you're interested in trying one:
http://editedmovies.ldsresources.net/#comparison
According to Clean Flicks FAQ they edit out profanity, graphic violence, nudity, and sexual language. So, they probably wouldn't be editing out "magic." In the end it is probably just like a regular movie that has been edited for the Pax channel.
Services like this also aren't all that new. I don't know if it was Clean Flicks but I know there was a service like this in the late '80s for VHS tapes. And in the early days of computer DVD players you could subscribe to a service that would download from the internet a table of content to skip when playing DVDs (so, if you were watching
Liar Liar it would know to skip four seconds at 23 minutes, 16 seconds in and so on).
I also don't know that I would blame these services on "American culture." I have now doubt they'd be in Europe as well except that the service provided violates EU copyright laws.