Around 1970-72 I recall seeing a movie on TV that was obviously
British-made (possibly a BBC production?), and it was about a rock
band that came up with a gimmick: they would play on the sympathies of
the audience to get them whipped into a frenzy, such as having the
lead singer in a cage on stage being prodded and beaten by Bobbies. In
the film, they were a big sensation. What is the name of this movie,
what year was it made, and is it listed on IMDB.com?

The film you are thinking of may be "This is Spinal Tap"

http://www.imdb.com/title/tt0088258/

Nope. Absolutely not "Spinal Tap."

What about "All You Need is Cash" a mockumentary shown on NBC in 1978
featuring a fictious band "The Rutles" who were a spoof Beatles.  See
http://www.rutles.org and http://rutlemania.org.  IMDB listing
http://www.imdb.com/title/tt0077147/

No, it's not The Rutles. I've seen that film, too. This was earlier,
probably made in the late 60s or early '70s. It wasn't a spoof like
The Rutles, it was more of a serious drama that seemed to be a
commentary on rock culture crazes.