I suspect the Sens will tease him out at center, perhaps even in the next month. But I don't think it's imminent.
He is still losing more than his share of 1-on-1 battles and figuring out when to challenge a D in the open ice versus use his mates, and he is also a little too cute in his own zone at times in trying to make a clever pass versus clearing the puck, but this is all about his development as an elite player. Smith has loved the way he has played overall, so he isn't going anywhere. Despite those parts of the game he's still figuring out, he is demonstrating night in and night out he is one of our best offensive players. He really should have a bushel more assists at this point given the setups he's made. He is playing well defensively, working hard and creating turnovers in the neutral zone consistently more than most other forwards. It's hard not to love watching him, even when he makes mistakes. Given his style and explosiveness with he puck, it's easy to overlook that his vision and touch passes might just be his best skill. You really need to watch him several games to get a full appreciation for that aspect of his game.
I suspect you'll see a few more trades by the Sens in the next month or so, including the trading of a center or two (Stepan, for example) at which point Stutzle will get some time at center before the end of the year. But I suspect a guy like Logan Brown will get a look at center first. For now, Stutzle is playing great as one of our top LWers and there is no urgent need to experiment with him at center yet. That will come.