Like C. Koca said, giants are mentioned to use weapons, both is the books and in GoT (for example, think of the battle against the army of the dead at Hardhome, in the episode called ... Hardhome).
So we got this fact : Giants use weapons, and, given their size and strength, a single giant with a tree thrunk would have destroyed the whole Bolton formation.
What I see is a scenaristic convenience. Of course, we can still search for reasons why Jon Snow might have chosen not to give him a tree trunk, but the series in itself doesn't provide one.
As we clearly see the same giant use a crude maul in the Hardhome battle, the series don't give elements about him being to stupid to distinguish his foes from his allies for example.
So my guess is : Jon Snow's situation had to be super desperate before the Vale's knights' intervention. The salvation had to come from them. The Bolton's formation was great at rendering this situation : the brave but unorganized wildlings were trapped, had no possibility to actually fight, were just being butchered.
Put a 4 or 5 meters hight giant waving a tree trunk in the equation, and their testudo formation is gone.
The giant could have died before the melee, killed by the cavalry, but he was needed as a convenient (and rather badass) way to break Winterfell's doors.
So the scenarists had to make his impact minor until this point. Not giving him a weapon, thus lowering his reach, was a great way to do this, while being less absurd than Jon Snow fighting without his sword for example.