Why should we avoid the traditional approach with non-linear models? Well, first of all, there are plenty of articles being published that still do mediation analysis in this way. A problem is nonetheless that the non-linear models are not directly comparable (due to rescaling between the models).  

There are different types of mediation analysis that one can employ to overcome rescaling bias – one of them is the KHB method (KHB stands for Karlson-Holm-Breen). Just remember that mediation analysis is only as good as your analytical model!

Mediation assumes causality, and while using proper mediation analysis is one step in the right direction, causal inference (see A note on causal inference) is still an issue if your study is based on observational data.