The assumption I need to challenge is that "an ultraintelligent machine could design even better machines".

That assumes that better machines are even possible, that there's no point of bottleneck that cannot be broken, or that they are so much more intelligent than us that they too won't hit a limit pretty quickly to what they can figure out.

It also says it surpasses any man however clever, but doesn't say it can surpass the totality of man interacting and building on each other's. If you take the most clever of all man, they are not so clever as to surpass humanity's total ingenuity and invention. And it's not a given that this super intelligent machine, more clever than any one man, is able to replicate itself and work in tandem with copies of itself in a way that is as effective as how humans somehow form a dynamic system between themselves.

Lastly, and I know the premise assumes it's better at everything, including at building intelligent machines, but I think an ultra-intelligent machine could exist that struggles at building better machines. Depending on your faith, we might be the construction of lesser intelligent organisms eventually discovering and evolving into us, and yet we do not yet understand how to make a more intelligent human.