Hey there,

I am really sorry for the delay of this reply as it is a weekend for us here.

Hm, that is no easier way to turn off this activity stream functionality completely.

But you can disable that story panel form from backend > Easysocial > user > profile type > edit your profile type > ACL > Story

And the group workflow, you have to add 'posting permission' field into this group workflow so group creator can able to set which user can able to post stream item into the group.