To put it into context I'm in a small team, our team is part of a larger 40+ team. There are things I would be comfortable saying openly with my team Vs the wider team.
Not that I have much of a filter anyway...
We tried doing a bimonthly quiz across our wider team but attendance waned and it was a lot of work for people to pull together content and manage etc.
In our smaller team, you are right and not everyone is into the idea every week, but they are vocal about it and the reasons (deadline/family etc). They are not penalised in any way shape or form for being honest.
We also had themed stand-ups for a while (it was good to cheer us up from time to time but a lot of work).
We also have to acknowledge that it's different for everyone, we all have different circumstances.
I have a daughter, two of my colleagues have two kids, one is single, one is in a relationship.
Half the team are from different countries, and have had visits home cancelled, they have been upset on calls I've been depressed about things too at times, but you have to be honest with your team, now more than ever you need to be supportive of eachother.
The most simple thing I've found is to have a variety of MS Teams backgrounds that are not your standard ones. It's a simple thing but can get a smile every now and then, especially when someone's kid appears!