What do the flat characters such as Polonius contribute to Hamlet by William Shakespeare?
It is somewhat of a misconception that fully rounded or developed characters are somehow better than flat characters. Actually, both have different purposes. A rounded character is important for literary works that are essentially psychological and where the main interest of the audience is the internal development or nature of a small group of characters. For other literary genres or emphases, such as moral conflict, plot development, and entertainment, flatter characterization can be as or more important.
Polonius is almostr a stock comic character who serves two purposes, plot development and comic relief. His character is that of a flatterer, who gravitates to power and who is constantly seen as currying favour. His death advances the plot of the play. In dialogue, he advances the discussion concerning Hamlet's madness:
Therefore, since brevity is the soul of wit
And tediousness the limbs and outward flourishes,
I will be brief. Your noble son is mad.