If I understand correctly, the new UI only allows custom grouping by the labels given in the Group tab. This implies that all alerts will be grouped by the same grouping label set.
The problem with that is that the alertmanager config allows different groupings for different alerts. That's the grouping that is relevant for sending out actual notifications. But this grouping is neither available in the UI, nor can it be re-created.
For a proper link-back from a notification to the alert group in AM, we need that grouping anyway.
If I'm missing any already existing UI element/feature, please let me know.