I prefer to pre-cook sausage before building a pizza because I don't want all that grease on the finished pie. Pre-cooking also means I don't have to take the done-ness of the sausage into account when determining when to pull the pizza.
Howdy David! Welcome, and I'm personally looking forward to seeing more of your posts this summer! From what you've shared so far, you're going to fit right in.
Your points do make sense, but I've never encountered either as a problem, in my own experience. We tend to buy our fresh Italian sausage from Caputo's grocery stores, and I've never had a problem with them being too greasy. I've also not had a problem getting the sausage to be fully cooked with the rest of the pie, though I suppose that would depend on the size of the crumbles. I've done both ways many times on oven-baked pizzas, and find that I prefer adding it uncooked. In my lone attempt at making thin, wood-fired sausage pies, this technique worked well there, too.
With the deep-dish, though, I am starting to side toward pre-cooking... Rendering the fat ahead of time makes more sense to me when the sausage is used below the cheese and tomatoes, and certainly makes sense in terms of doneness. You would only find out too late if the sausage wasn't fully cooked below the cheese and tomatoes.
I'm making pizzas on the grill on Thursday and was all set on wood-fired until Nate posted this awesome display of deliciousness. Now I'm wavering...
Edit: I prefer crumbled sausage to the solid sheet that's not uncommon in Chicago, but I am pretty certain those sheets of sausage are frozen when they go onto the pie. But whether they are pre-cooked or not, I don't know.