Specializations and many more can be unlocked by completing Service Assignments.
I'm assuming this is the objective you see in the lower right at start of game with the check boxes?
"Will you pick one kit and master it, or complete them all and become a jack of all trades?" ? ? ?
I don't see any way it lets you pick and choose, or skip one, it just gives you one, and you seem to have no choice in the matter.
Objectives need to unlock new weapons is spelled out in BF1 Companion, how hard would it been for the lazy people at DICE to add the new customization to Companion?
Then a while back, DICE removed all the weapon stats, damage, recoil, etc, not sure why they did this, it was useful to be able to compare. But with some many things, I can't begin to think what is going on in DICE's brain.
"Service Assignments are a new set of challenges that will test your mettle as a soldier. Show that you’ve got what it takes with the “Getting Started” assignments before moving on to the more difficult “Staying Focused” challenges. Will you pick one kit and master it, or complete them all and become a jack of all trades?"