S31 did help end the War, but they very nearly lost it too.
By slowly killing the founders they made a cornered, wounded animal of them. They made them more dangerous and desperate. The Breen came into the War, and who knows how many more would have come in if The Klingons hadn't been able to hold after the Dominion retook Chin'taka? Not to mentioned that if S31's plan had worked, the Jem'Hadar would have gone on a rampage lasting for what was it? A couple of weeks? Before they all died.
Oh, not necessarily.... I recall "The Ship", in which the Jem'Hadar force commited suicide en masse for failing to save the Founder.
Therefore...knowing that, I'd imagine 31 wagered the same thing would happen en masse were the Founders to die out thusly.
But those Jem'Hadar were still under the influence of the White.
Once the Founders are dead, there would be none left who know how to make the white. Those Augments Bashir helped came to the same conclusion. No white, Jem'Hadar go on rampage until they die out.
Except the white was being manufactured in the Alpha Quadrant, via factories referred to in that episode with the Jack Pack.
The Jem'Hadar would still have been Under The Influence when the founders died out--which would've been soon after WYLB.