Excellent read OP, very well thought out and informative.
In my opinion, I would say that the Spartan II's didn't show their faces because they were trained at a much younger age then the Spartan III's and the idea of being the best soldier in every way possible was more ingrained in them.
Spartans bring the idea of the perfect soldier to mind, strength not through individuality but through uniform warriors, so that the enemy doesn't see flawed individuals, but a powerful and (as long as the military power lasts) unbeatable entity (you knock one down and 2 more pop up).
I'm pretty sure somewhere in the books it says that almost no one can tell the difference between Spartans when the're fully armored, with the exception of Dr. Halsey and a few others that know them intimately.
The Spartan II's have accepted this idea, to not be remembered as people but soldiers. The biggest difference, for me, is that the Spartan II's have accepted it and the Spartan III's have not. The Spartan III's take off their helmets (Reach), and try to express themselves as people (Emile's helmet, Kat's arm).
You never see a Spartan II with anything non-regulation on or about them. They rarely if ever use sarcasm, joke or speak out of turn. That's what makes the Spartan II's stronger in my opinion, they aren't burdened with trying to express themselves, just completing their mission anyway possible.
This is all Speculative and opinion based....