Group ride Strava listed separately

Maybe your problem is a setting in Strava.  
Settings - Privacy Control - Group Activities.

No issue on this side. On the flyby of the activity, it appears that I'm a few seconds ahead of the other persons of the group. Probably linked to the timer of the Garmin after sleep mode.

I’m still having the same issue even with latest firmware. 
If I press ‘Stop’ at the cafe or press ‘Stop’ and then put to sleep, I will not be tagged in the group with any of my riding partners in Strava.  The only time it works is if I just let the unit Auto-Pause. 

Before Covid, I rode with a group 2 - 3 day per week.  We would meet up, ride down to the coast, stop at McDonald's for roughly 30 - 45 minutes, and then ride home.  I never bothered to turn off my Edge (older models, of course), and having the unit sit for that long never seemed to affect battery life by much.  If I were you, I'd just leave the 1050 turned on from the start of your ride to the end.  Then your coffee stop is just like a long red light!

Not worried about battery life. I have 3 ant+ connected lights and when I put it to sleep the lights all turn off. They reconnect and turn on when I wake up the 1050. I bring the 1050 usually into the cafe, if I have just let it Auto-Pause then the Garmin Varia Radar keeps beeping on my device when a car passes. I know I can just manually turn that off but at the end of the day the elapsed time still running is the issue here. 

I thought that was fixed. Let me try it again over the weekend

Sorry, I'm a daytime rider, and only use my lights very rarely. I didn't think about the issue of having your lights running!

(A few years ago, I was taking a medicine that made me sensitive to sunlight, and I bought the Varia headlight and tail lights so I could ride at 5AM.)