Why do we often use Jaumann rate instead of Truesdell rate?

In the attachment, we show that Truesdell rate can by simplified to Green-Naghdi rate by assuming F .=. R and can be further simplified to Jaumann rate by assuming W .=. R(.)R(T), where .=. means approximately equal

In a stretch dominant deformation, the three rates give different stress rate. This is usually explained by that we need a different tangential modulus for different objective rate. However, it is hard to understand why we need to change "material" modulus when we use a different "mathematical" form of objective rate as they are all supposed to be equivalent.

So a simple explanation may be that the Jaumann and Green-Naghdi rates are inaccurate when stretch deformation dominants. As we know that in a shear deformation, Jaumann rate gives a "sin" varying shear stress, while Truesdell gives the exact answer (See Ted Belytschko, Wing Kam Liu, and Brian Moran. Nonlinear Finite Elements for Continua and Structures.)  In another word, the three rate forms might not be equivalent, espeically when "stretch deformation dominants."

So a quick question is why are we so often using Jaumann rate, instead of Truesdell rate, in large deformation FE analysis?