#### Many econometricians/statisticians view the "force of growth" as the most natural measure and therefore see the percentage change as the approximation

The reason for this usage is that logarithmic difference gets you the "force ofgrowth" under continuously compounded growth, which is perhaps the most natural way to measure growth in a quantity over time.  (In different econometric contexts this may have a more specific name such as the [force of interest](https://en.wikipedia.org/wiki/Compound_interest#Continuous_compounding), etc.)  

If you have a nominal growth rate $r$ and you compound the growth $n$ times per time-unit for $t$ time units then the total growth $p_n$ over the time period satisfies:

$$1+p_n = \bigg( 1 + \frac{r}{n} \bigg)^{nt}.$$

Continuous compounding occurs when we take $n \rightarrow \infty$ which then gives:

$$1+p_\infty = \exp(rt),$$

and $r$ now represents the "force of growth".  One of the nice things about measuring using the force of growth is that it is comparable for things growing over different time periods.  The total force of growth over $t$ time-units is $rt$, which is a linear function of the amount of time elapsed so the rate $r$ has a natural meaning that is easily comparable across time periods of different lengths.  If you start with an amount $A_0$ at some time and then this grows to $A_t$ after $t$ time-units then you have $A_t = A_0 (1+p_\infty) = A_0 \exp(rt)$, which then gives:

$$rt = \log(A_t) - \log(A_0).$$

Suppose you have a series of values $A_0,...,A_k$ over times $0,t_1,...,t_k$.  One of the nice things about this result is that the total growth rate using the "force of growth" is additive:

$$r t_k = \log(A_k) - \log(A_0) = \sum_{i=1}^k [\log(A_i) - \log(A_{i-1})].$$

The fact that the total growth rate under the force of growth is additive and represents continuous compounding of growth makes it a natural measure for growth.  There are many mathematical benefits to framing growth in terms of the "force of growth" (continuously compounded) instead of the percentage change occurring over a particular period.  In particular, it leads to additive growth rates and it also covers the entire real number line without any lower bound.

Your question amounts to asking why we would look at $rt$ instead of looking at $p_n$, with the former asserted to be an approximation of the latter.  However, it is actually somewhat more natural to see the force of growth as the important measure, such that the percentage change in $p_n$ is actually the approximation to the more important quantity.