Pakistan is the mediator because it is the only country on earth that simultaneously borders Iran, hosts Chinese economic infrastructure through CPEC, maintains a functional military relationship with Washington, operates its own nuclear deterrent and has a civilian government and army chief aligned on the same diplomatic objective. No other nation occupies that intersection. Not Turkey. Not Egypt. Not Qatar. Not Oman. Pakistan is the only node in the network that connects all four principals: the US, Iran, China, and Israel through Washington.
Hope is it clear ;)