The various countries kept their best and brightest. You got the dregs because, let's face it: X-COM was probably meant as a sort of token international effort. No one expected you to succeed against the overwhelming alien force, so they sent you a few soldiers who at the very least knew which way to point a gun, while keeping their elite forces to defend against the inevitable invasion for as long as possible.
As much as you are trying to justify it with some context, let just face the reality. When the game was made, no such thing was given attention. They just wanted to make a game. Would the game have any challenge if you already started with Spetsnaz as your squad members? They don't even know how to use a medkit for gods sake.
Narrative consistency is fine but when it interferes with the gameplay, you need to forget about it. What game would you rather play? One which is consistent while sacrificing gameplay or one which is actually enjoyable to play rather than hear about?
As for your justification, it is loadsa bull crap.
The whole "elite forces" never come into play. They don't even have local police defending the citizens in a terror mission and no matter how long you wait the army never comes. The countries also prefer to sign pacts with the aliens instead of actually fighting them, leaving all the work to X-Com.