No, because (1) it isn't needed, (2) it's probably unfair competition to Verizon/AT&T/T-Mobile, and (3) it would be just as overpriced and delayed as SLS and F-35 (unless they contract it out to an existing carrier like FirstNet but that would have other problems).
Most major industries should be nationalized to an extent, healthcare especially, but also a nationalized rail system, electrical network, and yes, communication networks.
They should, but they won't. The closest thing we might get is deregulation, but meaningful deregulation is just as far away as the idea of a government-run network.