Thanks for your feedback, group by Column can be hidden but that leads to 2 problems:
1. You can't drag / drop that hidden column to the grouping toolbar. (which is obvious)
2. group by column title gets hidden when grouped rows are collapsed
It was not designed to be hidden as of now. Assuming you don't need the first feature when group by column is hidden, I've added the 2nd point to be considered in the future enhancements.
As for the UX is concerned, have you checked the merge cells feature? The group by column can be merged vertically so that it doesn't show repeated values.