While there's a lot to like in this movie, which is one of the best film adaptations of a broadway musical ever done, it has its troubling side. The central theme of both the show and the movie is, as the opening song indicates, the role of tradition in the lives of the villagers, and its message is that tradition must yield to newer ways.
The concept of "tradition" that is the theme of this show is apparently simply "the way things have always been done." They have no real connection with moral or ontological truths, and the attempts to answer questions about why traditional ways exist are played for laughs, indicating that they are not meant to be taken seriously. This kind of tradition (mere custom) is opposed to the new ways of doing things, which are equally unexamined.
So, one kind of mindless conformity (to the past) is simply replaced with another kind of mindless conformity (to the present), which is assumed without justification to be better or more practical. Now, if we are going to be mindlessly conformist, it's probably a lot better idea to stick mindlessly with what is tried and true than mindlessly to follow fads that are new and untried or, in the case of the practice of marriage based on infatuation, something that has been tried and found disastrous.
However, there is a better choice. We can try to base our choices on who we really are, who God really is, why we are here, where we are going, and what will help us to achieve our true purpose in life. This is an approach that does not seem to exist in the world of "Fiddler," and it is the reason I think the movie is charming but perhaps morally harmful. It encourages one of the most besetting vices of our age--a mindless reliance on what is new and what feels good to make choices that have profound consequences.