Why are there no complete problems for NP ∩ coNP?
It is still open whether $\mathrm{NP\cap coNP=QP}$, where $\mathrm{QP=\cup_k2^{log^k(n)}}$.
And in this answer, we have shown that $\mathrm{QP}$ has no complete problem.
If now someone shows that $\mathrm{NP\cap coNP}$ has some complete problem, that would imply $\mathrm{NP\cap coNP\neq QP}$.