I wouldn’t agree with “Tyrosh doesn’t have any significant land army we know of.” The Free Cities largely rely on mercenaries, but they have armies of their own, and they aren’t limited to hiring one mercenary company at a time.
Keep in mind that at the time of the War of Ninepenny Kings, the Band of Nine had conquered the whole of the Disputed Lands in addition to Tyrosh, so their power was not limited to the one city-state.