Your question seems to be when a variable is a ratio of two measurements each having uncertainty associated with it shouldn't we deal with the ratio like any other random variable? The answer is yes and this is often done. The delta method is one way to approximate the variance of a function of random variables when the variance(s) of the variable(s) that are inputs to the function are known.
In the case of ratios there are special issues. For example if X and Y are independent standard normal random variables X/Y is distributed as a Cauchy random variable which has an infinite variance and also an infinite mean. When the denominator has positive probability density around 0 the heavy tail can mean that the variance is infinite.
When you know the density functions for X and Y and they are independent you can write down the joint density of X and Y apply a change of variables mapping (X,Y) into (X/Y, Y) and integrating out Y.
Also by Jensen's inequality E(X/Y)>=E(X)/E(Y). The inequality is strict unless X/Y or Y have degenerate distributions. So there is a lot that can be done to deal with ratios of random variables.