Once again, LOOK AT A MAP! Then explain 1) why it would make geographical sense, & 2) how routing a pipeline through an additional country would add anything to it except more political risk.
A pipeline from Azerbaijan through Armenia to Turkey is logical geographically, but politics prevent it. Once that option is ruled out, there is no reason for any pipeline to pass through Armenia. It neither reduces the cost, nor reduces the political risk.
Ok I am jumping in from the middle so forgive me ... You are right.That is unless you are discounting Nigorno Karabagh as part of Armenia - which it is not, based on the demarcation done by Stalin & approved by UN when Azerbijan got its independence.
If you accept it then on geographic basis the pipeline, if it has to pass to Nigorno Karabagh, will go through Azerbijan, Armenia, again Azerbijan & then into Turkey.
But even if you accept Nigorno Karabagh as Armenian territory it won't change the political reality. Azerbiajn will never have any such deal with Armenia, especially which is based on the lands Armenia occupied from Azerbijan. Turks sharing ethnic morality with Azaris support Azarbijan & don't accept the occupied territories as part of Armenia.
So all goes into the garbage bin.
The only geographically & politically feasible rout is Azerbijan-Georgia-Turkey then into Europe.
Now since I have re-read your post; I would say you are completely right.