The system of Rock-paper-sciccors works very well in Fire Emblem, and this is how it would change over into Society. Sword beats axe, lance beats sword, axe beats lance. Archers are good at range but when the enemy starts advancing they loose effectiveness. Archers also are better agaist mounted units. As for siege units, they arn't in Fire Emblem but they could work as intensely good ranged units( or melee for rams) but if attacked by units and is undefended, it will be destroyed easily. Tell me what you think of that tasty goodness.

HELLO! Cantapults could totally own a group of infantry as long as they were in range.

If a catapult loaded up 50 1kg rocks instead of 1 50kg rock it would totaly destroy any troop formation it hit. And we must not forget the pots fired from catapults that were loaded with naptha/metal shards/other crazy death things. Catapults could make a real mess of any troops they fired on. But all it takes is one dude with a club next to it to smash the catapult to kindling.

On a side note, there should be many many kinds of units, so one guy could have roman liegon style stuff with boxes of spearmen with huge shields and longbowmen behind them and another guy who does the mongolian thing with vast armys of cavarly archers and another who gets a couple crazy berserkers with monster swords. There needs to be enough variety that it dosen't turn into a 3-4 stargaeys total thing.

The actual weapon isn't important, i was making a point that long range siege weapons like catapults are useless agianst you if your standing next to it. At that point you can foul the engine with a rock from the ground. After all, folks are not likley to stop and repair a catapult while folks try and kill the repeair crew.

The triangle system exists because in many situations it is true.

Spears will beat cavalry, this part is fairly obvious but it has the requirment that the spearmen be in formation and facing the cavalry charge, if they are scattered about or the cavalry hit from behind the spear is just as good as any other weapon agianst cavalry.

The idea that swords beat spears comes about because in single combat, if the swordsman and spearmen are both trained soldiers, the swordsman will almost always win. This is because the sword is easier to manuver and allows the swordsman to block the spear more quickly, which means he can move closer, and once the swordsman gets inside the spears effective range the spearman bassicly has a crappy club if he is swinging it, or a crappy dagger if he shifts his grip up closer to the head of the spear.
This was even more exxagerated in heavy fighting where there all the fromations were broken. A spearmen would have trouble swining his weapon around to face an enemy so, as you said, they would simply drop the spear and use a side arm, but once they drop the spear odds are they won't be picking it back up, so they might as well not even be considered spearmen anymore.
The only place where a spear would beat a sword is in large formations of spearmen where there is no gap in the spears for the swordsman to get inside the spears range. That kind of formation was effective agianst everything but ranged units. That and the fact that formations of spear wielders were easy to train and spears were cheap and easy to make is why the spear was effective until folks started using cannons.

On a side note you said any well trained spearmen could keep a swordsman from cutting the spear up, and while that is true a spearman was often very poorly trained, after all, all a spearmen has to do is march in line, and then stand in line holding is spear up and wait for someone to impale himself on it. That combined with the fact that many spearmen were just conscripted peasents meant that almost any swordsman would be vastly better trained than any spearmen he is likley to encounter.
The lack of training is actually why the swiss pikeman were so incredible. They were extremlly well trained and used formations so they could bassicaly massacare any other spearmen and could hold thier own agianst swordsman.


tl;dr - spears great in groups, crappy alone, catapults have minimum range