You keep going back and forth, yet you evade the question that has been asked several times.
Are you saying the EC is only responsible to set the long time trip function by turning the screw and then turning it over to others for testing, or are you saying the EC has a responsibility to verify that the breakers work in coordination with the others?
These are two totally different things! Regardless of what you think as an Electrical Engineer, many of us who are in the Contracting business resent the method your Company uses to take profit out of our work by buying the Switchgear yourself! Your Company could just as easily require sole source purchase. As such, unless you state in the specifications that I, as the installer, not the supplier, am responsible to perform testing to verify coordination of the breakers or testing to NETA standards, as a Contractor, I am going to fight you every inch of the way. As far as merely turning the adjustments, there isn't a NETA testing agency out there, including the manufacturer's personnel who would really care whether they were "adjusted" right before they came out or not, so I have to assume your Company is trying to force the EC to pay for NETA testing on Switchgear you supplied. So, I have to conclude that your Company IS asking the EC to pay for testing equipment you supplied or there really wouldn't be an issue.
Regarding what testing method is "most common", I think the reason you haven't gotten a satisfactory answer is because there isn't one. If it were on a job I estimated and Managed, it would go like this... If the Specifications required me to provide factory testing then the factory would do it. If the Specifications required, "NETA Testing" Then I would get an independent testing Agency (or two or three) quote(s) and if the manufacturer also provided a quote, and I would compare them using the least costly method that I was comfortable satisfied the specifications. No most common or least common, just least expensive. To which, I would of course put my mark up on. So it seems to me that you aren't asking the right question.