Exacly, there's so much to think about, because there are multiple perspectives.
Think about this. Why is fire emblem so unique? Is it because of the chess-like gameplay? The amazing stories and characters? The simple but outstanding rpg mechanics?
I think it got so unique because of the attachment to the characters. Everybody knows that every character is unique, stats/gameplay wise as story wise. This, right here, is why reclassing creates problems.
Think about a situation without reclassing. I got a thief. If the thief dies, there is no way for me to cheat chests or doors. I've lost a character, but also an invaluable asset to my team. The same goes for mages, bishops, knights, everything.
Now about a situation with reclassing. I got a thief. I know that swordfighter is the most efficient class. I reclass to the swordfighter, keeping the skills of the thief. I now built a character that can do everything a thief does, and can also has a better chance to win in battle. Do you see what kind of situation this creates, if this goes the same for all other characters?
I am significantly more willing to discard units that have low stat caps or growth rates (I can reclass anyway). I am more willing to let weak characters die because they can be replaced. I am capable of creating game-easifying (is this a word?) units. Causing me to think less about my gameplay decisions and causing the game to 'dumb down'.
Reclassing is overall a difficult element to implement and balance properly. It is a fun option because it gives more freedom to the player. But do we actually want that freedom in fire emblem?