My answer is plot, Virusbomb had a comment along those lines on another answer that made me smile. My 'in universe' answer can only be that he is a fool. First lets be clear that this is a skirmish force, not a large army deployed for battle using those types of tactics.
A single barrel of dragon glass could have made hundreds of
arrowheads. Based on the timing in the show they should have a few boatloads
ready, bringing back 50 arrowheads worth would have been no
Most skirmish archers are also very well trained at melee.
A decent archer can fire off a staggering amount of arrows at an
approaching enemy depending on visibility range. A great archer can
snipe at high value targets from a distance, during the battle
visibility cleared up. With low visibility high accuracy can be
achieved at 10 yards without fear of hitting your own team. Note that visibility cleared up during the big battle.
Given the experience and background of Jon I find the fact that he did not bring at least two bows almost beyond belief. Even if he did not expect to see any more than small groups of enemies, he would want range weapons.
I would love to hear of instances in real history where I would be mistaken. Where range weapons were available to any size force, on any mission and it was decided to stick to melee.