Sunfire cape is known as the wave pushing item for characters with little to no DPS.
You say that like it's true (AP amumu for example), but my point is that it doesn't have to be a pushing only item. if it were toggleable it would make more sense to buy it in other roles, leading to better item diversity and more meaningful choices in game.
Even if you're just split pushing, if an enemy champ walks up to you while you are damaging their turret, you draw turret aggro. that is probably not what you want. so sunfire isn't good if you're freezing, isn't good if you're pushing, and a toggle feature would fix both of these issues.