Why are ratios relevant, but not the increments when defining center frequency of a band-pass filter?
I don`t think it is correct to say that the center frequency of a bandpass "was defined like that" (based on the mentioned ratios). This property (geometric mean value of the 3dB-cutoffs) is only the RESULT of the definition. And the definition of a bandpass center frequency (second order) is simply based on the magnitude maximum at f=fo. This maximum is reached when the denominator assumes its minimum. In this case, both real parts of the denominator cancel each other and we have H(jwo)=Ajwo/Bjwo=A/B.

This is a real ratio. We therefore define: At the center frequency f=fo the second-order bandpass has maximum gain (minimum damping) and zero phase shift. 

This is the definition - all other properties are the result of this definition and can be used to make this definition descriptive and logical. 

EDIT: The explanations above are based on the classical ("normal") form of a second-order bandpass with a denominator written as second-order polynominal in s : H(s)=As/[1+Bs+(s/wp)²].