Does/Should the Gladiator background grant proficiency in the weapon it gives?

The Gladiator background (an Entertainer variant, from PHB p. 130-131) gives a strange weapon (like a trident or net) as starting equipment, but doesn’t explicitly specify that it gives proficiency with said weapon.

Is it implied that it does? (Or if not, should it, considering that a seasoned gladiator is expected to be particularly skilled with his weapon of choice?)