"Is the church growth movement a good example of Weber's rationalization?"

For your example to hold true to Weber's rationalization, the end goal would have to be to get people in the church doors, which the Growth Movement (Seeker Movement) is doing *exceedingly well*.  Although it is a part of the process, the real goal of the church is to lead people to Christ... and for that purpose I think the Growth Movement, without sound theology is largely ineffectual.

Has the church shot itself in the foot by compromising theology for the sake of popularity?  Yes, I believe it has.  It's the equivalent of "preaching another gospel."  Many of the churches in the midst of the seeker movement are all about Heaven on Earth, basically... Be good, feel good, and be wealthy.  You will be hard pressed to find this in biblical or historical Christianity.  "In this world you will have trouble" not "In this world you will have a three car garage."

If any church waters down the Gospel to make it more palatable they're taking a huge risk.  Even if the new visitors convert; the next generation of believers, fed all milk and no meat, will be completely unprepared  to offer even the most basic apologetic.

1Pe 3:15

but sanctify Christ as Lord in your hearts, always {being} ready to make a defense to everyone who asks you to give an account for the hope that is in you, yet with gentleness and reverence;

i go to church and see mostly older people.  if theology isn't presented in an up-to-date "fun" manner, young people will not go and your question will be moot.