I'm not sure I entirely understand but I think what your implying is that you believe there are both time-variant factors and country-level factors influencing your outcome variable.

The reason you don't see people in the literature testing for simultaneous influences is because in econometrics you often formulate a research question that seeks to find causality of one variable of interest.

Hopefully the paper I linked shows up. It seeks to find if one factor, having a national democracy, influences national GDP. The authors acknowledge there are country specific and time-variant factors that also influence GDP and seek to control for all of that to understand if their variable of interest (being democratic) has any causal impact on GDP. This is a two-way fixed effects models with a lagged dependent variable. Because of the inclusion of the lagged dependent variable they use a GMM estimator to correct for the serial correlation biasing the estimates.

https://www.aeaweb.org/conference/2015/retrieve.php?pdfid=333

I suggest you think about what your research question is because it is often less interesting to say "there are many things that influence GDP both within the country and across time." When you could say "after controlling for all country-level effects and time-variant effects it is evident that having a democratic government likely increases future GDP"