I actually thought about that last night and I agree. In musicals, people burst into song AND dance for little to no reason. But Les Miserables, the Phantom of the Opera, Sweeney Todd, these operas involve people relaying deeper emotions in a way that translates easier to its audience. No matter how good an actor's performance, many audiences cannot feel a true emotional attachment due a growth in genre savviness and a willing detachment from "fictional" worlds. Music, however, has always had a strong emotional impact on its audience. However, in most cases in Final Fantasy, the target audience cannot truly appreciate the music's emotions without stopping to just listen to it (like as a soundtrack). If the music played a role in the plot (as with the characters relaying some thoughts through song) then the player would be forced to absorb the music's relationship to the game and would be more greatly affected emotionally.

Some may mention that music from FF has affected them emotionally even during gameplay, but I expect they'll cite a scripted scenario like Aeris' death. But the mere fact that the scenario is scripted means you're actually doing anything other than pressing one button to move the text along. As I stated above, most people cannot be actively playing the game to appreciate the music as the gameplay distracts from that element.