Let me poke a hole in your arguement.
Archers are often characterized as calm, self-possessed, and analytical.
For your benefit I emphasized a key word in the description. Often. This means this is usually how it is done. That does not mean always. You could list hot headed archers if you wanted to. There is nothing on this page that would exclude them.
This little qualifier gives the trope more flexibility then say putting the word always in it or even just removing often.
If this were writen
Archers are characterized as calm, self-possessed, and analytical.
I would agree. But it is not. I find the claims of the trope being narrow or stiff to be an exaggeration.
This trope covers Elves.
In High Fantasy, elves usually share these traits, and are nearly always depicted as excellent archers.
Female archer characters.
The Chick getting stuck with a bow as a slightly more proactive take on the White Magician Girl. This also has the benefit of keeping the delicate female safely out of the bone-crunching melee.
Archers being conferred some respect for their skill.
Due to the amount of concentration and practice needed, archers usually receive a fair amount of respect for their skills.
Archers in a Modern setting
In modern settings, an archer is often a part of an ancient and noble warrior tradition, forsaking the newfangled vulgarity of firearms.
Archers as the loveable rogue. Robin Hood Expies abound even in comic books with characters like The Green Arrow
Loveable Rogue-type archers will usually have "Robin Hood" themed outfits, something like this◊.
A mention of archers in mythology.
One of the themes recurring across several mythologies is the proper archer passing the challenge with a strong bow, while the uninitiated can't even string the weapon. To understand why this is prohibitive, remember that the strongest bows are recurved ones and see the first photo here — if a man who did it all his life needs an assistant to do these gymnastics right, what are the chances of anyone unskilled, alone, against an exceptionally strong shaft when it's slightly dried and twisted?
Crossbowmen would fit nicely. William Tell comes to mind.
The Real Life Examples especially the mongols fit nicely.
I am not seeing how this trope is "narrow"
If I were to change anything it would be the bit about the Cold Sniper
and the statement that you can't "fire a bow".
The bits and pieces could probably use some touch up and maybe some rewording in general. I am not seeing a need at all for a rewrite, rename, or repurpose of this trope.