I've looked at design-build, but perhaps not in as much detail as I should.
Design-build is quite controversial. Architects and engineers don't like it. They think it gives far too much influence to contractors. They worry about the quality of the product because—and this applies to P3s—often if you have to redesign as you go, then usually you kind of dumb down the project and make it fairly generic and not very attractive.
On the other hand, I would argue that the main advantages of P3s are said to be coming in on time and on budget, which I would question some of the time. You could achieve that by design-build.
That's my only argument. I'm not saying design-build is the preferred way.