I believe it’s important to study and understand each approach in order to be able to apply an eclectic approach with the best out of different approaches.
For what I read, I agree most with TPR, CLIL, TBL and a bit of Discovery Guided approaches; but I do understand the need of Grammar approach because of the accuracy our students need to work on.
The approaches I found in my coursebook were lexical and communicative approach and a little bit of PPP.
About the Test teach test approach, I found theese links:
It’s very useful to have done a diagnosis, specially to intermediate or more advanced levels; so you’ll know where to start working.
By the way, I found this link with a summary of several approaches that might come handy