Cooked or raw mushrooms on pizza?
Short answer: It Depends.

Long Answer:

You are correct that the decision on precooking mushrooms before putting them on pizza is about controlling moisture.  For some hefty mushrooms like portabellos, it's also about making sure that they cook fully.  Here's what to factor in:


What's the density of mushrooms per pizza area?  That is, how much space between slices?
Are these hefty mushrooms (white, crimini, portabello) or delicate mushrooms (chantrelle, oyster, enoki)?
How thickly are they sliced?
Are there other "wet" items going on the pizza topping, like a sauce or fresh tomatoes?
How thick is the crust?
How long will the pizza cook, and how hot?


On one end of the scale, if you're making a thin-crust pizza with tomato sauce and covering it completely in thickly sliced portabellos, you pretty much certainly should precook them.  However, if you're making a deep dish pizza with only a few thin-sliced white mushrooms on top, then don't bother.  Stuff in between is up to your judgement.

For pizza restaurants, American Pizza is a medium crust, and they generally don't put that many mushrooms on each slice, so it's fine to put them on raw.  However, I've been to thin-crust Neopolitian-style pizza places where they did, indeed, partly cook the "wild mushrooms" ahead.

Also, if you are precooking the mushrooms, you want to cook them in a way that will remove moisture.  That means broiling them in the pizza oven on a baking sheet with parchment ... not sauteeing them.