Fully endogenous models for predicting multivariate time series
I am still interested in alternatives that depart more fundamentally from VAR, but digging a little deeper, I found some interesting variations around VAR that I did not know about and could be very useful.

Of particular interest are regularized VAR models. Applications to macroeconomic forecasting are for example discussed here: https://arxiv.org/abs/1508.07497.

Even better, the paper comes with a fully funcitonal R-package: https://cran.r-project.org/web/packages/BigVAR/index.html, that is described in more details here: https://arxiv.org/abs/1702.07094.

Some of the regularized method in the package, like VARX-L, allows for exogenous variables. As I described in my answer, this is not what I am directly interested in. But VARX-L (and the BigVar package) only allow for exogenous variables and do not require the inclusion of such variables (the exogenous channel can be shut down with the model still being operational). Good to know I can start with a fully endogenous model and have to flexibility to add exogenous variables later if needed.