I think that the part $(a)$ of proposition $2.4$ of this paper shows that for $n$ sufficiently large one can construct a $n \times n$ 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 one can take a unitary matrix $U'$ with $Det \;U'=1$ such that $0$ lies in the interior of the convex hull of the eigenvalues of $U$.
In fact if $U=ABC$ for positive $A,B,C$ then $UC^{-1}=AB$ so $C^{-1}=|AB|$ is the positive factor of $U $ in its polar decomposition. Of course such $U$ has $-1$ as an eigenvalue.