Touching Bow men, I persuade my selfe fiue hundred musketers are more seruicable than fifteene hundred bow-men; from that rate to the greater numbers in all manner of seruices : my reasons are thus, among 5000. Bowmen, you shall not find 1000 good Archers, I meane to shoot strong shoots; let them be in the field 3. or 4. months, hardly find of 5000. scarce 500. able to make any strong shootes.
And here is a summary of his arguments.
From my understanding of the question and the source material, Sir Williams need not be interpreted as referring to specific battles as evidence for the inferiority of bowmen in battle, but rather he speaks of the decline of bowmen in general, and thus exposes the weakness in the legendary bowmen of England: the laws and recruitment regimen that enabled them to harness the power of the longbow in their battles against the Scots and the French only gave them a temporary advantage, and the frequency of conflicts drained England of its capacity to field armies of archers in the 16th century as it did in the Hundred Years' War.
The prime of English archery was in the 15th century, with Agincourt (1415) being regarded as its greatest triumph. However, towards the end of the Hundred Years' War, England would suffer many losses that would impact on the availability of experienced archers: Patay (1429) was the antithesis to Agincourt, with the archers being caught out of position and slaughtered. Formigny (1450) negated the particular advantage of prepared positions that English archers preferred to fight from, with the French culverins forcing the English to battle, who in turn were counter-attacked. It has been said that the failure of the archers in these later battles was also the result of the decline of disciplined archers after decades of warfare.
Against the Scots, Flodden (1513) was one of the last deployments of archers in open battle, and by this point the armour used by the Scottish pikes largely negated the impact of the archers, as compared to early conflicts (Halidon Hill, Neville's Cross, Homildon Hill, Falkirk) in which the lesser-protected formations were vulnerable against archers. The last recorded use of the longbow in a pitched battle was in 1642 at Bridgnorth, in which a unit of longbowmen routed musketeers. This is point, however, where the mandatory upkeep of archers was largely ignored, and warfare had truly advanced to the era of gunpowder.
Sir Williams' argument is easily understood by those who practise archery today: the ability to shoot heavy warbows is rare and requires a lifelong dedication to the art, an approach that was well in decline for over 200 years by the 17th century. So when writers postulate on the usefulness of archers in the Napoleonic era, this is most definitely a sign of romanticisation rather than of practical benefit.