What you've described is in fact, exactly the same thing as log-odds - they're simply separated by a logarithm/exponentiation. Thus, all the multiplications you describe are the counterpart of the additions I describe. I agree, we could work with odds ratio, without taking the logarithm - but using logarithms has the benefit of linearizing the probability space. The distance between 1 L% and 5 L% is the same as the distance between 10 L% and 14 L%, but you wouldn't know it by looking at 2.72:1 and 150:1 versus 22,000:1 and 1,200,000:1.