Wear the sling in public because it keeps people from bumping into you. I'm wearing one now, but because I also have broken ribs I want to limited movement as much as I could, but I'm seldom actually using it. If you stay just three days in that sling, without ever moving your arm, you'll already be feeling atrophy in your muscles, really stiff painful joints. It's not worth the BS rehab that you'll go into IMO. On the other hand, with 3 cm separation wouldn't the bones need to stay relatively still in order to knit? I don't know about that, 3 cm is at the edge of "conservative treatment".
I have a separate sling that I use just for walking and running, where I added a strap at the elbow that reaches around my waist. It keeps the elbow fixed as a pivot point while allowing some movement of my arm, so far so good.