Why do we want the sum of the residuals to be as close to zero as possible?

zero residual equals zero mean

Step-by-step explanation:

Let me start by saying, unless expressed algebraically, this question would need a deep logical assumptions to paint it clearly and that is why the difference between data points and regression line is called "an error" or a residual.

To answer you, we want the sum of the residuals to be as close to zero as possible because:

zero residual equals zero mean and infact, if this isn't so, the entire value for mean isn't justified.

Best way out is through algebraic justification.