There’s a trope I’ve noticed with UML discussions and articles, which is that UML and agile are diametrically opposed.

This is not strictly the case. There’s nothing in UML or agile that says you cannot update a diagram in a sprint.

The supposed opposition comes from, not technical reasons, but the manner of thinking that each encourages. The promise of UML is that if you diagram the problem, all things will flow from it (implementation and documentation), this implies a design upfront and lends its self naturally to the waterfall approach, though this need not necessarily be the case.