The Bengals had already won the division at that point. If the Bills had lost and the Pats won in Miami, the Pats would won the division.
Joe Burrow was limping around at the end of that week 17 game against the Chiefs. Playing him in a meaningless game was have been reckless.
What the Bengals were trying to avoid was not the Pats, but rather the Colts. If week 18 had gone according to plan, the Colts would have been the 6 seed and played the 3 seed on the road.