In a single word, yes, more headroom is the reason.
But few things are really that simple. More watts can lead to better sound only if speakers were limited by fewer watts in the first place. Different speakers have different power requirements. You need more power only up to a certain threshold, a "good enough" point. Beyond "good enough", more is not better.
Similarly, different listener's preferences and different room acoustic properties can affect this. And last, but not least, is price. The added cost of increased amplifier power is not small. And one man's price ceiling is another man's floor.
Another way of looking at the question of the value of increased amplifier power is to ask yourself this. Given the cost of roughly tripling the amplifier power, how much could I improve sound quality by instead spending that same amount of money on better quality speakers? I hope you get my overall point.