I think it is less important how the Iconian gate works, and more important in what its effects are.
They can move things but what things? If they're cheap and ubiquitous, how would Iconian society be different structurally? Do they need ships?
Ships move people, materials, goods, and weapons. The gates obviously move people, so ships are not needed for that.
Raw materials are possibly needed, but they could be extracted with ease by shoving a gate aperture around underground anywhere in the galaxy. Each refinery type, if different types are even needed, could each have their own gate swinging around under ground and inside asteroids just scrapping out continuous projections of door shaped materials. For capturing gasses from gas giants, or liquids from oceans, use a gate with a regular size opening in the refinery, and a giant opening on the collecting end.
What about finished goods? It's questionable finished goods need shipping if the Iconians have replicators as advanced as their gates. They would only need raw materials as feed stock. However, if they have straight energy to matter conversion, they would have vice versa, and the whole idea of refining anything is pointless. They could run everything off anything. That also assumes they don't have some sort of perpetual motion machine as their power source, and they would probably have one or something very close to one.
But, let's assume they do need to move finished goods. They don't need space ships at all, instead have a conveyor belt, arm, or tractor beam feed a rapidly fluctuating gate, and send all items directly to where they need to go, with no middle process. Every time an order goes though the gate, it changes location, and the faster an item clears the gate, the faster it can switch to the next location.
That leaves weapons. Again the Iconians don't need ships to delivery destruction to a target. They can build their weapons on planets, and put a gate at the muzzle, and fire the weapon into the gate, sending the attack anywhere in the universe. Unlike a ship they would have the power of an entire world, or entire multi-planet civilization if all power grids are gate linked, powering their weapons.
The weakness is easy to imagine, such a system would make it very easy to over centralize all things on the homeworld, even though it would be just as easy to over decentralize. Let's assume they were bombed to death only because they had the psychology which lead to over centralization. But why put gates on other worlds, as seen in DS9? It might indicate gates really are only one way devices and that a colony does require a return gate. Although, why bother at all with a near completely centralized society? Maybe it was a vacation spot in the wild, since the Iconian homeworld would likely be completely urbanized.
The one thing I didn't like about the DS9 ep "To the Death", is that the supposed Iconian technology seen there has little resemblance to what was originally seen in TNG's "Contagion". Is this another ancient relic, or a newer more advanced one? Is this evidence that the Iconians survived, or is it another abandoned artifact?
I would just accept it as an older gate used for a different purpose. I has a much messier effect than the TNG gate, but other than the inrush of gas like energy it is the same as the TNG gate in its special effects.
