All classes should be able to use all weapons.
Marksmanship principles are mostly the same across all weapons platforms, with only some small differences between pistols and assault rifles, and crew-served machine guns have some slight differences as well. There are no differences at all between assault rifles and submachine guns. Even with some of those small differences, if you've got the fundamentals down for one it isn't hard to learn how to use another, since some principles are exactly the same no matter what weapon type you're using.
Finally if the character is military or part of some paramilitary organization, he or she should be trained in all small arms and machine guns in that organization's arsenal. An Alliance Marine, no matter the class, should know how to use a pistol, a submachine gun, an assault rifle, or an LMG like the Revenant. From a lore perspective it doesn't really make sense that certain classes wouldn't know how to use certain weapon types.
That said, for gameplay reasons there has to be some differences between classes otherwise there would be no point in playing a pure combat class like the Soldier. On that note I think Bioware should bring back weight restrictions. Allow the soldier to carry all weapons types,, whereas biotics and the like would be limited to one primary weapon and one secondary. (an assault rifle and a pistol for example).
The only exception perhaps would be bringing back heavy weapons but maybe having them a soldier class exclusive. A biotic not being able to use heavy weapons could jive with the lore, where the ability to use biotics is heavy linked to minimizing fatigue. Loadout weight would be a big issue for a biotic even from a lore perspective.