Let me put it this way: In an even battle, the first edge you gain is by far the most important.

The defeat of Ba Sing Se would end all resistance to the fire nation because the Earth nation would be defeated, and so the Fire Nation would gain a decisive advantage by controlling a large continent. Eventually, the Northern and Southern water tribes would fall, simply because the Fire Nation has and had so many more resources at it's disposal. It's a lot easier to fight one enemy than it is to fight two.

Of course, one would imagine that a more compentent general be in place, and the point where the drill pokes through the walls would be surrounded by earthbenders plugging up the wall and attacking the drill, probably winning (because the last major earth city should be protected by more than enough to stop a lone large Fire Nation battering ram).

Yeah, I think about this stuff a lot.