"Their best players were injured!" - False
Injuries can hurt a team heavily, especially if key players get injuries. The Tampa Bay Rays and Texas Rangers have lost a large chunk of their starting rotations this year, yet they are both still contending for division titles.
Spotrac provides total number of days on the injured list for each team. While these data points include long-term injuries that teams expected to happen (for example, surgery at the end of the 2022 season), information can still be gathered that is reliable.
Per Spotrac, the Cardinals rank 23rd in total days with players on the injured list, meaning 22 teams have had more "injured days" than the Cardinals. While other playoff teams have experienced fewer injuries such as the Diamondbacks, Phillies, Astros, Blue Jays, and Orioles, there are still plenty of playoff teams that have performed well despite facing huge amounts of injuries. The Dodgers, Reds, Twins, and Braves have all experienced more than 1,500 days on the injured list. The Cardinals cannot blame injuries for their down year.