Yes, but the only characters in Discovery we were invested in were Burnham, Saru, Tyler and Lorca (and to a lesser extent, Stamets and Tilly). Of these, only Lorca is really deeply invested in the Klingon War, the rest are more engaged in day-to-day shipboard operations or the task of trying to work out the spore drive. As it stands, the ethical and social dilemmas surrounding Burnham's fall from grace -- and, relatedly, the implications of using the Tardigrade to run the spore drive -- VASTLY eclipsed the relevance of the Klingon War itself.Wouldn't that emotional connection depend on the personal investment in the characters?
I think it's worth remembering here that Discovery isn't actually about the Klingon War. Discovery is about Burnham and Saru and the crew of the Discovery. The war serves as a plot device to add a sense of urgency to the development of the spore drive while also raising the stakes for the usual Alien of the Week plot device (it's always the same alien, and their encounters are part of a wider conflict). If they had written the war out of Discovery, then the central storylines could have been exactly the same, except the stakes would be a lot lower for the characters and for the viewers.