Makes one wonder whether sending those 100 infant changelings to the rest of the galaxy was such a smart move (assuming conquering the rest of the galaxy already was their plan back when they sent them out).
If one looks at the Dominion War story-arc as a whole, their curiosity about Odo was their undoing.
My guess is their justification for their actions in "The Search" was that Odo was trying to come home. They seemed to have done something that gave him an urge to come home. He was motivated to go on the mission, and was being drawn to the Omarian Nebula the moment they crossed into the Gamma Quadrant. "No Changeling has harmed another." Would denying a Changeling the ability to come back to the Great Link be considered "harm" among their community?
It's their need to connect with Odo that betrays their existence as the Founders.
It's their failure to understand why Odo remains at DS9 with Kira and the crew, and why Odo defends them against other Changelings, that leads to them punishing him, and infecting the entire Great Link with the Section 31 virus.
And in the end, it's Odo going home that causes their surrender.
The Alpha Quadrant might never even have had a clue as to what was going on until it was far too late...
I've always assumed that's what happened in the alternate timeline of "
The Visitor."
There's been no war with the Dominion in that timeline. Since I don't think the show would want people to think removing Sisko from the timeline leads to a kind-of peace, I always assumed the Klingons being given control of Deep Space Nine leads to the Martok-Changeling opening the door wide, and all the major powers have basically been neutered.
If you come at the episode from that perspective, it does give more purpose to "Old" Jake's obsession with bringing Sisko back. Leading a "normal" life was never possible, since Jake was really living in a dystopia controlled by the Dominion without anyone knowing it.