Archery was actually rather undervalued in our real history, particularly in medieval Europe, so it isn't terribly surprising for it to be undervalued in the relatively close parallels of Westeros, which was deliberately based in many ways on medieval Europe.
In the feudal systems of Europe, knights, the backbone of medieval armies, were the elite nobility who could afford heavy armor and good steel weapons.
Archery was relegated to low paid professional soldiers or peasant conscripts for centuries. Many knights disdained archery as "beneath their station" or "unchivalric".
It wasn't until relatively late in the history of feudal Europe that longbows became recognized as an important strategic asset in warfare.
One of the most famous early battles where longbows featured heavily was the Battle of Agincourt. During the battle, the English forces were outnumbered somewhere between 4-3 and 6-1. However, the vast majority of the English forces (5/6th of the total troops) were archer units.
This was seen by many as a crippling shortcoming at the time, as evaluations of comparative strength were frequently based on the number of armed knights on either side.
In fact, one account overestimated the already severe imbalance of troop ratios up to "ten French nobles against one English", dismissing the archers altogether.
Yet the archers were incredibly decisive during that battle, resulting in thousands of deaths and prisoners on the French side, and scant hundreds of English casualties (possibly as few as 113).
One of the problems with archery in feudal society was that the relatively heavy armor used by the knighted nobility required use of heavy longbows in order for archery to be effective in piercing the targets.
Archers required extensive training in order for these heavy war longbows to become effective in warfare scenarios.
Given this, it makes sense that most of the use of bows you list involve targets who are primarily unarmored (wildlings, victims of the Ironborn raids), or are used by those who may not be able to afford steel (countrymen and rebels) or who employ hit-and-run tactics (rebels).
Bran and Arya, and other members of the high nobility, likely trained primarily for hunting and for tournaments (although knowing Ned Stark, it is entirely possible he wanted them proficient for more practical reasons).