I want to look at the impact of a number of factors (e.g nutrition, alcohol sales, GDP, etc.) on mortality. I have a dataset with 10-30 years of data on these variables aggregated at the state level. (i.e. I'm looking at the impact of total state-wide alcohol sales in a given year on total deaths in that state in that year). Obviously, population size is a likely omitted variable if I don't include it. However, I am wondering whether I should include it by transforming variables to their per-capita form or whether I should control for population directly.

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.

What should I do here? Are these two approaches functionally equivalent?