The French, upon arriving on the field of Agincourt, actually did wait one night, precisely for the reasons you describe. Henry V was extremely cautious the entire night of October 24 to guard against being taken by surprise from a sudden French assault, but nothing happened aside from a minor skirmish or two. On the next day, however, both armies lined up for combat, yet neither actually advanced. Henry gave a risky order to advance the English lines forward and replant their stakes within arrow range of the French. They managed to do so without being attacked, and began to pepper the leading ranks of the French with arrows. This prompted the French to immediately begin the assault. Although arrow volleys at the most extreme range possible had little impact on them in terms of actual casualties, no knight worth his horse could bear the insult of an enemy attacking him without offering a response. Thus, they initiated the battle that would result in so many of them being slaughtered on a muddy field.

Before the battle actually began, as French forces were stalking the English around the countryside, the French had actually developed a pretty thought-out plan that incorporated their own archers, as well as a special cavalry unit whose job was to hunt and destroy the English archer units on the flanks. However, this plan had to be modified when more and more French nobles arrived with their own men. The order of battle shifted as they all wanted to be part of the units that would actually fight the enemy. In the shuffle, the crossbowmen and other French archers had gotten shifted from a forward position to one behind the ranks of French men-at-arms. It seems that the French thought that the missile troops might get in the way of the general advance, as they had at Crecy in 1346. Frankly, I'm not convinced that they would have made altogether much difference. The contribution of French archery to the Hundred Years War is minor at best and laughable at worst. Skirmishing against English archers was generally not a winning proposition, even for professional Genoese crossbowmen or experienced Spanish skirmishers.

The overall English strategy on the march from Harfleur to Calais was intended to force a battle with the French. In that sense, Henry's goals in 1415 were very similar to those of Edward III in 1346, during the campaign that resulted in the Battle of Crecy. True, the English were outnumbered, but they were a tight-knit, disciplined, and experienced fighting force. Overall, Henry's army was led by men with an enormous amount of combat experience. I'm not only talking about the aristocrats and knights, but also the archers. Many of the regular archers were on their first campaign, but their leadership and upper ranks were drawn from the same pool of men who had been fighting for the crown in Welsh and Scottish campaigns for a long time (EDIT: for clarification, "long time" here means about 10-20 years before Agincourt). Henry had every reason to imagine that he could defeat a numerically superior French force. The French were forced into a position that Henry had set up and forced to attack by Henry's tactical decisions. They were simply outmaneuvered by an extremely capable and cunning commander.

They wanted ransoms from captured English noblemen, and it made a number of them very impatient. It also prevented them from acting cohesively, since you really aren't going to give a share of whatever ransom you get to the guy next to you--this was every man for himself. The French also completely disregarded the English archers, thinking them no real threat.

It's also important to note that, when the English redeployed, the French forces were in range of the English archers (barely). They could have retreated, yes, but that was not likely on their minds. They had the numbers on their side, and waiting was also not to their notion of warfare.

The French camp was very disorganized overall--when they began their cavalry charge, a large number of men weren't even ready for it--some had wandered off to warm themselves, or otherwise made themselves unready. After that disaster, they pretty much had to advance. And, by the time they realized that was probably a big mistake, there was nothing to do but continue--they couldn't really retreat, because of the press of men from behind them, they were wounded and exhausted, and going back would mean they'd either have to walk backwards, which was a bad idea, or they'd have to turn around, which was another bad idea.

Also I would be interested to hear why the French ranks seemed to eschew the use of their crossbowmen in the battle, apart from an initial volley at the beginning.

Mismanagement. They were ill-deployed initially, and were unable to really be brought to bear after the advance of the cavalry and infantry. The narrowness of the field made it difficult to put them into place.

I have been told that there was a downpour before the battle and that the crossbowmen were probably hampered by their strings stretching where as the longbow archers were able to unstring their bows then restring them before firing. Anyone able to confirm\deny?