The use of semirelative sensitivity for \$I_C\$ on \$V_{BE}\$?
"Sensitivity for Ic on Vbe" ?

The sensitivity of Ic with respect to (changes of) Vbe is the most important parameter of a BJT - it is identical with the slope of the transfer characteristic Ic=f(Vbe) - and this is the so called "transconductance gm".

However, this parameter gm has nothing to do with something like a "quality factor". In contrary, we are interested in large values of gm because it determines primarily the voltage gain of an amplifier stage.

The quality of a bias network is a measure of the "stiffness" of the bias condition with respect to unwanted variations of surrounding influences like temperature, power supply, current gain hfe,...

As an example, the following expression defines the sensitivity of the collector current caused by variations of the exponential Ic=f(Vbe) curve only (Ico and hfe constant). 

That means; We are NOT interested in the slope of the exponential curve (gm=dIc/dVbe) but in the change of the slope caused by variations of the environment. These variations can be expressed by unwanted gm changes caused by Ic variations and - therefore, we need Sx=d(gm)/d(Ic)=1/Vt because of gm=Ic/Vt (Vt=temperature voltage).

Using the definition of gm=d(Ic)/d(Vbe) we, therefore, can write: 

gm=d(Ic)/d(Vbe)=Ic/Vt and Sx=1/Vt=[d(Ic)/d(Vbe)]/Ic

Note that the stability factor Sx as defined above shows only the dependence of the transfer characteristic Ic=f(Vbe) with respect to the temperature voltage Vt - expressed by the corresponding Ic variations (and NOT yet the more relevant influence of Ico on temperature).