I don't think the Klingon war on DS9 was shown in that greater detail than the one on Discovery. For episodes focusing on the war, we had the opening battle, a sneaky undercover operation to end it diplomatically, and a cease fire violation after that. Other than this, the war was never really a focus of the plot and was mostly used as a backdrop: Rules of Engagement had a false flag operation intended to discredit a Klingon Starfleet officer as a setup for a courtroom drama episode, Sons of Mogh was an exploration of Worf's family against a backdrop of the Klingons trying to mine Bajoran space, and other than that, we only saw the official declaration of war in Broken Link and the signing of a peace treaty in By Inferno's Light. I'd even say DS9 was worse than Discovery in this regard, as it quite openly showed life going on more or less the same as usual despite the war supposedly raging on.
But these points aside, Discovery had its own mission related to the Spore Drive. It was primarily a research vessel, not a front-line warship. The first season wasn't about the big battles of the Klingon war, it was about the role of Burnham and the Discovery in it.
Yes, looking at the specific episodes it was probably wrong to say the DS9 war had 'more' screentime, when it really had 'about the same' screentime, which is still not praise since the DS9 war was a minor hiccup that the Federation barely noticed and ended very quickly while the DSC one lasted months and supposedly nearly destroyed the Federation. This is also why I don't think life going on is much of a criticism of that DS9 war arc, because it was never supposed to be a life-altering war in the first place. They did that story later on with the Dominion and did it quite well overall, imo, though the semi-episodic requirements of the show taxed things now and then.
I agree Discovery had its own mission and this is why we didn't get to see much of the war, but that's not much of an excuse for me. The pilot episode(s) and the finale arc both very much make the war and the very idea of human/klingon conflict the thematic heart of the season, with the secondary theme being the struggle for the soul of the Federation - a struggle that explicitly only gains traction because of the supposed prolonged devastation of the war. That just isn't supported by what was shown in the middle, precisely because the Discovery spent far too much time by itself, away from the war, not dealing directly with the klingons, and not dealing directly with the devastation they left in their wake. Not even dealing directly with the Federation's inner demons, because Burnham's outburst is over already in the pilot, Landry dies almost instantly, Ash's conflict is forced and artificial and Lorca - the obvious avatar of the any means necessary mentality - was never really starfleet and never actually gave a damn about the war.