Each class should be able to use every weapon decently (as in be able to pick it up and shoot with it), but they should each get some sort of proficiency bonus that encourages the use of that class's weaponry.
For example, the sniper class should be able to start with precision weapons like the DMR (make it live up to its name?) and also lay down when using a sniper rifle or enhanced binoculars when using a non-scoped weapon like a Plasma Pistol. But, their movement speed should decrease when holding a heavy weapon. A demolition class player can use a sniper rifle, at the cost of accuracy and making it very difficult to score a no-scope.
But if Halo starts going down that road, it may turn into Halo Team Fortress instead of Halo 5. I don't really want to have something like a Master Heavy instead of Master Chief.
I say yes, but each class should have active gameplay differences instead of a little perk that you have the option to not use at all. I bet no one even uses the Stalker's Nemesis support upgrade.