DS9 is a great show but like I say it is a completion/companion piece. DS9 only sticks out so much in peoples minds because of the strong contrast it has with TNG.
On TNG we saw what life was like in the post-Undiscovered Country galaxy. With peace with the Klingons and the StarFleet primarily focused on exploration and adventures in the final frontier. There were some curveballs thrown at TNG, the Borg in season 2, 3 and 4, the Klingon Civil War, Romulan-Vulcan Unification fiasco, the Cardassian incident in Chain of Command, contact with super beings like Q, the Traveler and the alien from 'the Surviors', and time travel fiascos. However it was all apart of the journey into the frontier and daily life was comfortable in the Federation outside of that.
DS9 with the introduction of the Dominion threat, a coup d'état attempt by a bent StarFleet admiral, the Maqui-Cardassian Fiasco and Section 31's genocide scheme you have more ridged show. Tough problems require tough decisions. However I don't think this means DS9 is a more competent show than TNG, just that the environment was crafted in a way that caused the DS9 crew to respond the way they did.
DS9 is Gotham to TNG's Metropolis. Gotham is dark, hard, corrupt, and needs hard solutions to fix it's bent function. Metropolis is light and bright, but monsters still fester there and strike out at it. Each has the appropriate hero to deal with the problems of their respective city. Could Superman fix Gotham? Sure, just like Picard and crew could navigate solutions for DS9's problems. Would Batman's tough tactics work in Metropolis? Of course only the variety of crime is different. There is no competency threshold DS9 has sole claim to. It's purely environmental.
A lot of people cite "In The Pale Moonlight" when Sisko uses says he can live with the consequences of ordering an assassination of a Romulan delegate to coerce the Romulan empire in to the war, and how Picard and Kirk could never do those things. No one ever seems to see how this risky game could've blown up in Sisko's face and led to and even swifter defeat of the Federation by the combined forces of the Dominion and the Romulans. Compare Sisko to Anakin Skywalker for a second. Both of them committed acts of evil in order to serve their own agendas. Sisko with the assassination to preserve the Federation and Anakin assisting the execution of every Jedi. However both maintained they could live with their actions because they would make up for it later. In summation "committing small acts of evil now but making up for it with greater acts of good later". Sisko's was the preservation of the Federation and peace in the Alpha and Beta Quads despite the potentially tens of thousands of Romulan lives that would be lost. Anakin's was saving his wife and children using the force to ressurect and save people from death, at the cost of the Jedi across the galaxy. Sisko's gamble turned out in his favor. The war was won and he never had to live with the consequences of his decisions for long since he soon departed this reality to live with the Prophets. After seeing Cardassia decimated, and learning the full scope of what the war cost in lives, Im sure Sisko wouldn't have remained "Defiant" about his statement about how he could live with his decision to get the Romulans into the war. Anakin's gamble didn't work out in his favor. He lost his wife and children, was mutilated by his mentor and friend and was burned alive. He was placed in a living life support suit and struggles to breath with his seared lungs, and never gained the powers he was promised.
You have to think what if. What if Sisko's gambit hadn't worked in his favor? Would people be singing his praises for his bold decision that worked in his favor?
DS9 is a great show, and easily in the top 3 of the Trek series. Is it the best or most influential? I'm afraid not.