Isn't that kind of what already happened? BioWare looked hard at Skyrim and DA:I tried to be an open world game while cutting back on plot and character development, meanwhile Bethesda's Fallout 4 now has a voiced protagonist, dialogue wheel, and romanceable companions 
I don't mean them taking influences from each other; if anything, I mean the opposite. Have Bioware make a solidly Bioware game with their formula and their style - set classes to choose from, no real open world, a limited number of party members you take with you all the time and develop in depth relationships with, and just set it in Tamriel, using the tools available to you there.
Then have Bethesda make a very Bethesda game; one large open world area, less restriction on classes(would still need some due to the way mages work, but who says a mage can't learn how to pick up a sword), a largely solo adventure but with more, less developed companions, where you only get one at a time, and have it set in Thedas with Thedas' rules.
Just thought it might be an interesting experiment for them, to play with someone else's toys, but their own rules.
Anyway, in regards to Weekes. I do like him; in fact, by the numbers he may be my favorite Bioware writer, having written most of my favorite ME characters. However, he's 1-2 for me with his Dragon Age characters as I was profoundly meh on Solas and hate Cole. I do love Bull, though.