I came across this seminal paper on SABR model where the value of the call option is computed (eq. A.52).
After the dollar value of the option has been derived, a lot of effort is being put to express the (exact) dollar value of the option in terms of the implied volatility in normal (Bachelier) model (A.56) and then using this value to deduce the implied volatility in the lognormal (Black) model (A.65). The implied volatilities are then expressed as small-volatility expansion in terms of $\varepsilon$ which forces us to accept an approximation up to a given order and results in complex terms akin to (A.65).
The authors state:
This yields the option price under the SABR model, but the resulting formulas are awkward and not very useful. To cast the results in a more usable form [...] we find the “implied normal volatility” of the option under the SABR model.
It's not obvious to me why do we need to take the exact solution and transform it into an approximation in a different model. There are numerous papers concerned with implied volatilities that apply sophisticated techniques like singular perturbation, heat-kernel expansion or Laplace method in large deviations framework, so can someone please explain to me why it's worth the effort?