True. But the crossbow didn't have as much range as the longbow, and the French knights were trying to close the gap and overrun the English longbowmen. That is when the mud and their weight bogged them down and they were slaughtered.
The entire French "plan" was stupid. They had thousands of infantry that they shoved into the back of their lines and who basically sat out the fight. Those guys could have been sent through the woods to engage the English left flank. The crossbowman could have been pushed forward to compensate for their lesser range. If nothing else, it would have forced the English bowmen to divide their attention and their arrows. Instead, they just charged straight up the middle (three times, I think).
Please excuse my 20/20 hind-sight generalship. I swear, I just don't understand how the English lost the 100 Years War...