I agree completely with the premise that Tillman should be far more acclimated this season than last. It is hard to join a contending team mid season and fit in, get comfortable, right away. Just ask Derrick White. Tillman also had some personal issue (I don't recall exactly what they were) and he was coming off injury. I expect Tillman to be better, more productive for the Celtics this season.
As to how he fits in, I don't think being the 1 big, playing with Tatum as the next biggest is good for Tillman. I like him playing opposite Horford. It allows Horford to focus more on scoring and passing, Tillman can defend and rebound. To start the season, I think Kornet may start, and play as the single big, with Tillman and Horford playing more together. Horford of course will play with the starters to close games.
When Porzingis is back, I see him being the starter of course, so Kornet maybe doesn't play as much and Tillman's role may not change all that much. Maybe less minutes, but in largely the same role.