I think that the part $(a)$ of proposition $2.4$ of this paper shows that for $n$ sufficiently large one can construct a unitary matrix $U=-I_{2}\oplus U'$, which can be decomposed as the product of three positive matrices. For such $U$ we have $\parallel U-I\parallel=2$.
In fact if $U=ABC$ for positive $A,B,C$ then $UC^{-1}=AB$ so $C^{-1}=|AB|$ is the positive part of the unitary decomposition. Such $U$ has $-1$ as an eigenvalue.