BuddyPress.org

You should be able to create a page called Activity and go to your wp >> reading >> settings and change it to static page and choose your activity page. that should put your activity stream up on the front page.

but if you really want something like the twitters as you say..then those are not really live tweets happening. they are editorially run through a continous loop like the plugin you provided. thats what I think is what you’d want to do. I’ve been looking into the same thing for a while and it turns out it’s better to run the most recent Xnumber of activity updates in a scrolling activity loop so it looks active. running the activity refresh isn’t necessarily going to give you that same effect on the homepage if you don’t have constant activity. the closest to reaching this is the activity bar plugin, it just needs to display more than one activity at a time to accomplish this.

Here’s some other related threads:
https://buddypress.org/community/groups/creating-extending/forum/topic/scrolling-activity/
https://buddypress.org/community/groups/buddypress-activity-stream-bar/forum/topic/scrolling-activity-display/

Good luck. looking forward to seeing this hit the stage.