A constant that seems to come up in the 14th century is that armies did a relatively poor job of establishing sweeping supply lines. Even as far back as the Crusades, foraging and plunder seemed to make up a significant part of how medieval armies kept themselves supplied in the field. Yet supply trains and lines were understood by Alexander and the Romans.
At Crecy and Agincourt, Edward and Henry respectively had been relying on plunder to keep their armies afloat, which is probably why you saw less significant, large scale, armed clashes in the western Europe during this era.
What was it about the culture, political scene, or region that saw such sloppy management of supply lines for armies?
Awesome question, thanks for asking! I'll do my best to answer though my focus area is specifically medieval Britain so perhaps another user can answer in a way pertaining to continental Europe.
The first thing to consider when approaching medieval supply situations, especially when you compare them to Rome and Alexander, is to understand the massive difference in scale, both of actual resources available as well as men, expertise, authority, etc. While you're right that a early Imperial Roman Legion would be well supplied on the march, you also have to realize that the entirety of the Roman infrastructure and supply organization was available to them. They could be resupplied via land or sea at hundred of different ports, towns and cities. A 13th century English army on the other hand, would A) Be much smaller normally than a Republican or Imperial Roman force, B) Have less resources, money and transportation to call upon and C) Be significantly less centralized and organized all together than the Roman or Alexandrian example. So really comparing the two and saying "well these guys were first and they had good organization, why don't these later dudes?" presents significant difficulty.
The key element, in my eyes at least, that contributed to inefficient and sometimes non-existent supply lines is the decentralising effect of feudalism. Now feudalism is itself a controversial term nowadays with historians, and there were no societies that were perfectly "feudal," but in this context I'm talking about the gradual devolution of political, military and economic power to landholding aristocracy as opposed to having it in the hand of one specific ruler and his court.
Now by the timeframe you're talking about (Hundred Years War) significant centralization had occurred, thus that Edward I and his son and grandson could call upon more organized military machines than their predecessors. But, as you point out, there was much to be desired. Though, I would posit that in some cases Medieval rulers in fact performed admirably in terms of their supply administration. Edward I, for example, made use of the ships of the Cinque Ports (Ports who owed feudal service in the form of ships to the King) to resupply his troops in Northern Wales in the late 1200s. This supply network, as well as his efforts to maintain a road link to his base at Chester (Edward's army chopped their way through many a Welsh forest in their efforts to do so) allowed Edward to actually winter his army in Northern Wales instead of withdrawing, something few medieval rulers were capable of doing. He would replicate this feat in Scotland during his later wars there. The point is that Edward and his aides were capable in 1283 of keeping a significant force in the field over a prolonged period of time, and this eventually led to victory.
When it comes to your comments surrounding the English kings in the Hundred Years War, it is true that the English to some extent relied on raiding and pillaging to sustain their army in the field. But at the same time they had a dedicated supply of arrows and other essentials coming from England all the time, thus allowing them to expend thousands of said arrows at any given battle. The "living off the land" you mention was also an intentional strategy to bring the French out of their castles and fortresses, and was successful in doing so.
I would also point out, in conclusion, that this is not a strictly medieval mode of operation. Napoleon's armies in Europe were widely known for their proclivity to live off the land, allowing them to outstrip their supply lines and move faster and lighter when on the move. By the same token, Napoleon also ran into severe supply difficulties when his armies moved further away from their bases, such as during his invasion of Russia. Heck, even the belligerents in WWII experienced severe difficulties once they moved beyond their friendly zones. Supply problems are endemic to all armies, and many throughout history have been inadequate at solving them.
Davies, R. R. The Age of Conquest: Wales, 1063-1415. Oxford: Oxford UP, 1992.
Morris, J. E. The Welsh Wars of Edward I: A Contribution to Mediaeval History, Based on Original Documents. London: Oxford UP, 1968.
Bachrach, David S. "The Military Administration of England: The Royal Artillery (1216-1272)." The Journal of Military History 68, no. 4 (2004): 1083-104.
Prestwich, Michael (1972). War, Politics and Finance under Edward I. London, UK
Prestwich, Michael (2003). The Three Edwards: War and State in England, 1272–1377 (2nd ed.). London, UK