Yes, Web Designer allows text on a circle or on a path.

What Web Designer does not allow is Convert to Editable Shapes (why not is beyond my ability to comprehend). You can Combine Shapes > Add Shapes which arguably does the same thing, except you cannot separate the circle from the converted text. There is no Break Shapes.

More frustrating in Web Designer there is no Baseline Shift which could raise the text enough off the circle than when Add Shapes was applied, the circle could be removed with the Shape Tool.