What does the term "simple" in the phrase "simple whole number ratio" mean in Gay Lussac's law?
3/2 is simple. 45498334/9232493 is not.

The reason it works is because reactions of gases producing gases involve relatively simple compounds. If they were complicated, the molecular weight would likely be high enough that the substance would not be a gas.

This is akin to Loong's comment about the Law of Multiple Proportions, which also mentions "small whole number ratios." The difference is that for the Law of Multiple Proportions, "small" ratios doesn't apply in the case of something like palytoxin ($\ce{C129H223N3O54}$).

But palytoxin isn't a gas. For the reason mentioned above, you're almost certain to get simple coefficients for gas to gas reactions.