I'm not sure we can argue the Dominion was preparing for "war". After all, they themselves claimed they had lived in peace for either 2,000 or 10,000 years - that is, their expansion in that time had not involved major wars, just some easy crushing.
Now, the Dominion may exaggerate the stability of their reign, but its length is likely to be close to the figures quoted. It establishes a rate of expansion, and suggests that Earth wouldn't be in risk of assimilation yet for a couple of thousand years at least. And probably much longer, since the bigger the Dominion gets, the less distance is added between the center and the border for any given annexation of a volume of space, for simple geometrical reasons.
Whether the Dominion would have been a threat to the Federation when the two met through conventional means is difficult to tell. The UFP seems to be making rapid technological advances, such as being on the verge of transwarp in the 23rd-24th century already. The Dominion isn't noted for such, and might become obsolete in just a couple of centuries.
Then again, perhaps the UFP will soon stagnate, having found it impossible to actually do transwarp, and the Dominion is what a stagnated empire looks like, a couple of millennia into its existence. The two might indeed slug it out eventually on more or less equal footing.
The war we saw was certainly made possible by the actions of the Alpha players: the Dominion would supposedly not have found the wormhole without them, and would not have learned of the Alpha civilizations any time soon. And Sisko personally was pretty much the one responsible for the opening of the wormhole for traffic. But the subsequent Alpha push into Gamma, and the provocation of the Dominion into doing something about it (even when they had plenty of oppressing and conquering left to do in Gamma, and could simply close the wormhole if they so wished), was a group effort, and while Starfleet was in it big time, it was not Sisko's job to prod into Gamma. And it was not Sisko who first met and provoked the Dominion, supposedly, but one of those starships declared missing - UFP or Klingon or perhaps other.
Technically, Sisko again personally launched the war in "Call to Arms", by mining shut the wormhole, but it was clearly a preplanned action because it involved the synchronized launching of armadas of thousands of starships... And it just put a date to an event that was going to happen anyway.
Timo Saloniemi