Unless you're referring to specific historiography, for the most part the sources are pretty consistent on the effectiveness of English archery. Historical sources and modern historians corroborate on events in which English archers played a decisive role (Falkirk, Homildon Hill, Crecy, Agincourt) and when they didn't (Patay). There may be some bloating of the capabilities of the English archer by contemporary commentators (such as Prince Louis Napoleon claiming that a skilled English archer would loose 12 arrows per minute, while modern historians and archers are much more conservative with 6 to 10 shots). Otherwise, what happened as recorded in chronicles basically happened. When the English archers were superior, they were very much superior.

The issue I think you are identifying is today's pop history and history enthusiasts simplifying the effectiveness of English archery, particularly in overrated the penetration of a war bow and citing high draw weighs (150lbs+) as the reasons behind English archery dominance. This overlooks the social, economic and military changes that made such a force possible:

• The English citizens were required to own and train with bows, as stated in the Assize of Arms 1252 and subsequent laws
• England imported very large amounts of resources for bowstaves and arrows
• English archers were a semi-professional standing force that was maintained while other troops were levied
• English armies during the Hundred Years War and the Scottish Wars of Independence has disproportionately high numbers of archers
• Archers were deployed very effectively in defensive positions, or were deployed against stationary defensive formations with impunity (against Scottish schiltrons)

This goes alongside the battle effectiveness of the English bow, but also remember that there were lots of archers in English armies (possibly ~7000 at Agincourt, compared to ~2000 men-at-arms).

I don't think there are too many sources which claim that English archers were that amazing, given that they suffered equally bad losses and England ultimately did not win the Hundred Years' War with their archers. Longbowmen would continue to be used during the Wars of the Roses and even the English Civil War.

In short, I think the problem is that today, history fans praise the English longbow in the same vein as the Japanese katana, citing Agincourt as the exemplar of medieval archery, without fully understanding the social and military context of the period.

Further Reading:

• Donald Featherstone, The Bowmen of England
• Robert Hardy, Longbow: A Social and Military Context
• Mike Loades, The Longbow