The general public didn't need to know exactly what the Dominion was on TV, only that the condition of war existed already, and the story would have been self-contained but on a larger canvas. I feel the interest of the three-tiered enemy, it's ships and tech and greater inherent gravity in-universe would be apparent.
Look at notable war movies and tv series' of our era. Here is a sample; War Horse, Saving Private Ryan, Black Hawk Down, Platoon, Full Metal Jacket, The Longest Day, Band of Brothers and the Pacific. How many of them go in to specific detail about why each conflict was started and how many of them show a resolution to the overall war between nations in their depictions? Not many and yet the audiences are never lost and each movie reaches financial and critical acclaim.
For something like Star Trek a padded, hand holding opening like what LOTR Fellowship of the Ring did. Spending the first 10 minutes laying down the foundation of what the conflict was about with narration and depiction could've worked. When the bulk of your target audience (general not the fans) isn't familiar with the material (i.e haven't read the LOTR books, or watch DS9). An opening showing the audience what is going on between the Fed and the Dom could've worked. You really only need to paint a picture of the Dom as the antagonist since the Fed and Ent-E crew is already known as the protagonists. Similar to other war films, TNG vs Dominion film doesnt have to end with the resolution of the entire war. It can just be an adventure drama that climaxes with a grand battle like most war films.