I'm deeply uncomfortable with this as well. I've struggled with my mental health and had days where I could barely drag myself into the office. If this were implemented at my company, I would have spent the better part of a month beginning these meetings with 'well, I'm experiencing near-constant despair.'
Afraid of guarding? My mental state is my own business. I may choose to share it with my manager or my coworkers (in fact, I have), but to have it become a required part of a meeting? No. No. No. Especially on a remote team where I just don't know the other people. This is not the right way to build rapport.
It's telling that so many of the comments on this post at the time of my writing are negative. Please heed them.
And please keep in mind that as the team lead, or manager, or whatever your role is, there is a power imbalance. People will not have an easy time saying no to this activity, even if they want to.