I just don't get why the activity cannot support group submissions.
It's not intentional. It's just the feature has not been implemented yet. There is a feature request open to add it - MDL-35594.
Is there a way of getting an "unofficial support"?
Sorry, I guess "official" was not a good English word describing what I meant. I meant that the feature has not been implemented in standard Moodle distribution. Some institutions have their custom variant of the Workshop that adds it - see the issue above for more details.