If I have a member variable inside the MyView that stores the text, and create a public setter for it, then just calling that method causes the MyView to redraw itself
Setting a variable inside the View will not invoke a draw on the View. In fact, neither does the view system know nor care about internal variables.
Invoking invalidate() on a View causes it to draw itself via the View. You should check this out: http://developer.android.com/guide/topics/ui/custom-components.html.
A TextView internally invalidates itself when you invoke setText() and redraws itself with the new text set via the setText() call. You should also do something similar.