A member doing the identical activity — running, the same lifts, the same class format — day after day is at real risk of an overuse injury that cross-training would have prevented.
Different movement patterns distribute the stress
Mixing in a different training mode periodically gives overworked tissues a break while still maintaining overall fitness.
It's prevention, not just variety for its own sake
Framing cross-training as injury prevention, rather than just 'mixing things up,' tends to make members who've settled into one routine more willing to actually try it.