As mentioned in the previous discussion, it is difficult to answer this question in a generalized way since it depends on what your specific research question is. As far as adding a categorical variable for state, this is a separate effect. It may control for population, but is also going to be controlling for tons of other things such as different laws, cultures, incomes, etc. that can vary by state. On the topic of controlling for confounders with ratios, I'll go into some more detail than the previous discussion that may help a bit. You gave an example in your previous discussion:
It seems that creating per capita variables is the conventional thing to do, but it seems to me that just adding in population size as a control is simpler and it allows for population size to have its own unique effect on the dependent variable. For example, in this case, we might think that states with larger populations, all else being equal, have more deaths because it is harder to organize the healthcare system efficiently to meet the demands of a large population.
It's really important in studies where you're using a regression analysis to know exactly what your response is and exactly what your covariate is. I'll try to give some more input into when ratio makes sense versus population.
If you are concerned that absolute population itself is an important factor, as in your above quoted example, then directly using population as a control would be sensible. You might want to transform population to make it more manageable, but it makes sense to your study. In medical studies, it would be said that using this covariate "makes clinical sense."
Using per capita is a rate. It does not measure population or control for the same (or even similar) effects. For example, measuring absolute alcohol consumption and measuring absolute population and measuring alcohol consumption per capita are all totally different measurements. None of them are right or wrong, they just depend on your study.
Going back to your previous discussion, the response variable was mortality. Now, it is highly unlikely that absolute alcohol consumption is particularly meaningful if you don't know the population. It's practically impossible to interpret meaningfully. You could add absolute population as well. Then yes, population is controlled, but it is included as a main effect. In other words, your regression would be something like $$mortality=alcohol + population$$ But in this model, there is no interaction effect. The model is separately controlling alcohol and population. It is NOT going to take into account the same thing as alcohol consumption per capita. It is going to check and see "how does alcohol consumption independently impact mortality? And also, how does population independently impact mortality?" If your model includes a per capita consumption, the model is including what is actually implicitly an interaction effect. To make this clear, instead of writing $$mortality = alcoholpercapita$$ you could also for conceptual understanding write $$mortality = alcohol\times \frac1{population}$$ Which is going to check and see "how does the rate at which the population consumes alcohol impact mortality?"
I hope that describing it in terms of addition versus multiplication makes the ramifications a bit more clear. And maybe now you see that it's really impossible to know which you should pick as a control without knowing the specific research question that you're trying to answer.
At the risk of being verbose, here's an example to try to make it more concrete. Pretend you have the following data:
Nebraska has a mortality of 5, alcohol sales of 10, and population of 10, while Kansas has a mortality of 1, alcohol sales of 10, and population of 1.
If you control separately for alcohol sales and population, your regression equation is going to estimate that alcohol sales have no impact whatsoever on mortality, but it will estimate that mortality greatly increases with population. After all, alcohol sales are the same between states, but Nebraska has a 10 times larger population and a 5 times greater mortality. This regression says alcohol is irrelevant.
If you control for alcohol sales per capita, your regression is going to estimate that mortality increases with alcohol sales per capita. The practical translation of this is the more alcohol being purchased per person, the more people are dying. And this is probably what you're after in such a study.
