Why standard errors in country-level variables are higher than that in firm-level variables?

Consider a global panel model where there are many firms M >> N, the number of countries. If we are expansive about what we mean by firms to include small businesses, sole-proprietorships, and freelancers, then all domestic economic aggregates are going to be the sum of the firm level aggregates (employment is the sum of domestic firm employment, national output is the sum of domestic firm employment, and so on). Aggregate variability of these statistics is going to be less than the firm level variability. In THE GRANULAR ORIGINS OF AGGREGATE FLUCTUATIONS BY XAVIER GABAIX, he shows that if firms are identical and have the output variance $sigma^2$ (variance as percent deviations from the mean), then the GDP standard deviation (of percentage deviations from the mean) is $$sigma_{gdp}=frac{sigma}{sqrt{n}} $$

This going to make GDP much less variable than the average firm and approximately zero. An insight of his wonderful paper is that variability in firm size matters a great deal, and in practice: $$sigma_{gdp}=frac{sigma}{log{n}} $$ This is much, much larger, but still $sigma_{gdp} << sigma$.

Of course, you asked about the standard errors and not the standard deviations. Recall that the basic calculation of the standard error of the mean calculated from independent and identically distributed random variables is: $$ SE = frac{sigma}{sqrt{T}}$$, where T is the number of observations. While things get much more complicated as we allow for heteroskedasticity, serial correlation, clustering, and the like, the basic idea remains that you need more observations to shrink the standard errors. Since we established that $sigma_{gdp} << sigma$, if we observe firms and nations the same number of times (T) and the observations are IID:

$$SE_{gdp}=frac{sigma_{gdp}}{sqrt{T}} << frac{sigma}{sqrt{T}} = SE_{i}$$

That's my argument for why (generally) the standard errors at the national level should not be larger than at the firm level. I'm 99% sure you can cook up examples that flip this result with the right covariance structure. Nevertheless, it does not have to be the case that standard errors of country level variables are larger than firm level ones, and in the simplest case, the opposite is true.