The Properties Manager panel shows the “raw” representation, the dialog shows calculated value for radius. The latter is what an end user would more often need, as you said, but the software uses coordinate points internally which are always exact. Having an exact radius and exact coordinate points are contradictory and something must be calculated for a rounded value.
A circle is more simple than an arc. It’s a known and universal problem that arcs may have more than two representations and different kind of applications must either choose one or have several kinds of arcs, for example two endpoints and a third point, or two endpoints and radius, or start point, radius and angle.