First off, weapon and armor tiers:
Tier 0: clubs and staffs (what you start the game with), no armor.
Tier 1: Spears/Short Spears, Daggers, Axes, Warhammers, leather armor, wooden shields.
Tier 2: Boar/Winged Spears, Shortswords, Broadswords, Maces, Battle Axes, Chainmail or Light Plate, Kite Shields
Tier 3: Pikes/Short Pikes, Longswords, Greatswords, Mauls, Masterwork Chainmail or Plate, Tower Shields
Against leather-armored opponents, just about any weapon of a given tier is about as good as any other weapon in that tier (exception: daggers - never use these against anything if you can avoid it, that +4 initiative isn't worth dealing half the damage an axe would, though it might be on par with the warhammer due to the low initiative).
Against any form of chain armor, you're better off with piercing or blunt weapons than with cutting weapons.
Against any form of plate armor, all weapons of the same tier as the armor perform more or less equally again.
The real test comes in how long you can make use of a given weapon. The basic spear will outperform any similar-tier weapon against most later game armors, and will perform about as well as broadswords against any form of chain (and better than shortswords), despite these weapons being a tier better than the basic spear. Basic spears will never outperform maces, but will outperform warhammers against anything superior to chainmail (this includes light plate, since light plate is better against blunt attacks). Thus, by merely researching Training, you've gained access to a weapon whose maximum damage will only degrade from 4.9 against leather to 1.8 against plate and a tower shield. Other weapons degrade much more rapidly - broadswords can deal at most 2.88 damage against a chainmail and kite shield armored unit, while basic spears can deal up to 2.72 damage to that same unit, and cost far less. If you face masterwork chainmail-armored units, basic spears actually outperform broadswords (maximum damage is 2.45 for spears against unshielded MW Chain-armored units, while for broadswords maximum damage is 2.25). Against plate, broadswords fare somewhat better, having a maximum damage of about 2.77 against plate and a tower shield as compared to basic spears' maximum of 1.81 (boar/winged spears, however, have a maximum damage against this target of 3.90, and maces can deal up to 2.91, despite plate having twice as much defense against blunt attacks as against cutting, though you'll make more attacks with the broadsword than with either the mace or the boar spear).
Thus, for trained melee units, I would recommend using spears, because:
1. There are no armors in the game currently that specialize against piercing attacks
2. The damage capabilities don't degrade as much as with normal weapons over the course of the game
3. You aren't risking a powerful counter-attack with each attack your troops make
4. Spear-type weapons generally perform at least as well as cutting weapons against plate armors, and at least as well as blunt weapons against chain armors (accounting for the initiative penalty of blunt weapons).
5. Archers don't act that frequently, and bow damage is pitiful against armored targets (for reference, Yew Longbows, the best bow not unlocked by the 'Archers' faction trait that you can give a trained unit, has a maximum damage ranging from 1.33 (full plate and a tower shield) to 1.88 (masterwork chain without a shield) against late game armors, while against the same armors the basic spear has a maximum damage ranging from 1.81 (full plate and a tower shield) to 2.45 (masterwork chain without a shield)). Since you can deal roughly 50% more damage with a beginning of the game spear than you can with an end of the game bow, why are we concerned about arrow fire? Granted, against start-of-the-game armors, the damage balance is reversed, but if you've researched up to Yew Longbows you should have been able to research at least the Boar Spear, which can out-damage the Yew Longbow against Leather Armor by roughly 20%, and the relative performance only gets better for boar spears with improving armors.
So, yes, in my opinion there is little reason to bother with any weapons other than spears for trained troops, especially since spears are among the lightest weapons in the game (most one-handed swords have the same weight, but you'll probably also have a shield with those, and they may be exposed to counter-attacks, one-handed blunt weapons and the axe are heavier, all two handed weapons that are not spears weigh at least twice what spears do).