From the description of the spell:
The target becomes a Skeleton if you chose bones or a Zombie if you chose a corpse (the DM has the creature's game statistics).
"[T]he creature's game statistics" includes the actions shortsword and shortbow. However, unless the skeleton has a shortsword /shortbow (and arrows) this is an action that is not available to them. They would need to be provided with these weapons.
In the absence of these, they can use the Attack action to make an unarmed strike. If you give them different weapons they can use the Attack action to attack with those, however, the DM would need to decide if they were proficient with them.
As for your assertion that a skeleton has an "an unlimited-ammo shortbow" - this is just wrong. From the Monster Manual p. 11:
A monster carries enough ammunition to make its
ranged attacks. You can assume that a monster has 2d4
pieces of ammunition for a thrown weapon attack, and
2d10 pieces of ammunition for a projectile weapon such
as a bow or crossbow.