You are correct in everything you have said or suspect about using oil in a dough.
Not all doughs call for the use of oil (or other form of fat). A good example of this is Neapolitan style doughs based on the use of 00 flours. But I wouldn't recommend cutting the oil out of the NY style dough, such as the Lehmann dough formulation. The oil in that formulation serves many useful purposes, including the following: 1) it makes the dough easier to stretch (improved extensibility) by lubricating the gluten and starch molecules; 2) it captures and holds the nice flavors released during baking, and also contributes its own flavor components; 4) it helps to provide a better rise to the dough during baking, as it traps gasses; 5) it produces a good mouthfeel when the crust is eaten; 6) it contributes to the browning of the crust, through heat transfer (and oxidation) at high temperatures; and 7) when used on top of the dough, it provides a barrier to migration of liquids from the sauce into the dough, thereby preventing or minimizing a gum line (which is a no-no for professional pizza operators).
There is nothing that says that you have to use olive oil, or any particular form of olive oil. I use a mild olive oil such as the Bertolli Classico olive oil or the Carapelli olive oil, but you can also use a pomace oil (the lowest quality olive oil) or even a salad oil, grapeseed oil, a canola oil, or a blend of olive oil and canola oil (Tom Lehmann himself often advocates a 20%/80% olive oil/canola oil blend as a lower cost alternative to olive oil all by itself). Except for the real pizza purists, most professional pizza operators will usually go for the lowest cost alternatives. I would tend to reserve the highest quality olive oils, such as the cold-pressed extra-virgin olive oils, for other applications, although I will sometimes use it on a freshly-baked pizza for an added flavor component.