The docs link you shared is not of prophet handling exogenous variables its handling holidays which is a separate "feature".

Nevertheless, prophet's exogenous influence impact/explainability is bad. One other problem with Prophet's regressor (exog features) functionality is that say you have 10 exog vars. You'll have to go through every possible combination of the 10 vars to come up with the best one. This is exponentially increasing compute.

On the other hand ML algorithms are nice for this reason, if you do data pre-processing right and take care of multicollinearity and endogeneity to some extent, influence of exog is much more explainable.

As someone mentioned m5 competition. Do check this out, you'll find a lot of reasons as to why ML based approaches that learn on panel data are SOTA right now. Do not skip experimentations tho.