Why use the income approach to approximate GDP if it relies on the value produced by the expenditures approach?
The definition of the expenditure measure of GDP is:
$$GDP_{EXP} \equiv C + I_g + G + X_n$$

The definition of the expenditure income measure of GDP is:
$$GDP_{income} \equiv I_n - I_f - C_c$$

As I understand it, $s$ is not part of it. $s$ is defined as the difference between the two measures:
$$s \equiv GDP_{EXP} - GDP_{income}$$

Apparently it is also possible to define it with the RIPSAW approach:
$$GDP_{income} \equiv R + I + P + S + A + W$$
R = rents, I = interests, P = profits, SA = statistical adjustments (corporate income taxes, dividends, undistributed corporate profits, and W = wages.

And yes, it is useful to calculate GDP with multiple methods. Among other things, it reduces noise in GDP estimates:


  Two often-divergent U.S. GDP estimates are available, a widely-used
  expenditure-side version $GDP_E$ , and a much less widely-used
  income-side version $GDP_I$ . We propose and explore a “forecast
  combination” approach to combining them. We then put the theory to
  work, producing a superior combined estimate of GDP growth for the
  U.S., $GDP_C$ . We compare $GDP_C$ to $GDP_E$ and $GDP_I$ , with particular
  attention to behavior over the business cycle. We discuss several
  variations and extensions


Improving U.S. GDP Measurement: A Forecast Combination Perspective Aruoba, Diebold, Nalewaik, Schorfheide, and Song (2012, ungated here)