Yes he is eligible. No one is talking about him because most people were not expecting him to start the season in STL and because there is a lot of rookies expected to do better than Schwartz. There was an article on TSN a couple days ago that had schwartz about 20th among rookies
People also don't know what to make of Schwartz in general. He's not your typical high octane prospect. He has nice hands, but he's not fancy. He's elusive, but he's not a speedster. He wins battles and goes to the net, but he's on the smaller side. He hasn't lit up a professional league like many of the most anticipated rookies have.
The only way to know what Schwartz provides is to watch him play a game instead of just highlights. And then plenty of the people who have watched him see his size and don't grasp his cerebral, detail-oriented game; so they don't come away with the excitement that a flashier, but worse prospect can provide.