Allow me to niptick a bit : militia should be a term reserved for urban and semi-rural infantry of medieval armies.
Yes, formally you are right but I met the term "feudal militia" used in the meaning of "arierban", assembly of king's vassals, with an implication that the discipline in such an army was quite low and so were its tactical capacities.
The problem of the french army was, at the contrary, its particularily important diversity (most of the troops were footmen) which alone wouldn't have been a problem, without unability to enforce a single unified tactical decision, and unability of Philippe to really dominate the war council.
IIRC, at Crecy bulk of the troops (or at least those who saw the action) were knights with their bands. Unfortunately, most of the numbers related to the French army at Crecy are fantastic to a degree that does not worthy of a serious (or any) discussion. It is not even quite clear if its numbers had been greater than those of English army. " Whether Phillip IV was able to assemble an army of the same size or even stronger during the six weeks following Edward's raiding in Normandy on 12 July is not known. Those troops that previously had been fighting in Cascony and were marching up as fast as possible had not yet arrived. ... Phillip's decision to accept battle would still be quite understandable, since he was certainly superior in the number of knights." (Delbruck) His initial decision was to postpone the attack until the next day but the things got out of hands due to the enthusiasm of his troops. "If there had first been an orderly deployment and the entire mass had then charged simultaneously against the English, the English arrows would hardly be able to stop an assault. But the French moved up in the individual actions, just as they arrived on the battlefield with one charge following another ..." Probably a better general than Philip could manage to keep things under control but the task would not be trivial.
That the leading part of the army was made of feudal recruitment wasn't really the problem (Jean II really gave it priority in campaign, and in spite of Poitiers' catastrophic defeat, it was essentially a sound idea politically and militarily), but Phillipe's poor skills on this matter IMO.
The sound idea would be to have the mercenary bands. Their social composition would not matter but, almost by the definition, the heavy armored cavalry assumed the "knights" no matter how recruited. While not being what we now mean by the "regular army", these bands had at least some discipline and the trusted/competent leaders capable of controlling them.
John's problem at Poitier was a combination of his low qualifications as a general, low discipline (rush cavalry charge just due to a personal quarrel between Connetable and Marshal) and bad implementation of a seemingly sound tactical idea: charging English position by the big infantry columns (which would be something similar to the ill-understood Scottish tactics that won at Bannockburn but without the long spears, with the improvised "infantry" being overburdened with the heavy armor and with no previous training and with the enemy in a better position and better led). Of course, the dismounted French knights (at Poitier) did not make a good infantry: they completely lacked needed experience of acting as an infantry formation (training IS needed) and their infantry lances cut short were inadequate weapons. And, of course, they had no idea how to repel a cavalry charge so that they easily panicked (IIRC, one of the columns fled even without being seriously attacked). But, let's be realistic, John was a brave knight, not a capable commander.