Dear Alison Reeve,
I'm sorry to hear that SharePoint group is not working when you try to pick a member in the list column.
I also tested in a SharePoint list. I'd like to convey that I reproduced the same situation as you described, it just shows No Result Found if I tried to pick a member from SharePoint group. Very sorry for the inconvenience caused.
Then I did further tests and found that I can pick member from SharePoint group only when I manually add the members into the Site Members group with steps below:
From List Settings > Permissions for this list > click the Site Members group, open it, click New from ribbon > Add users> enter the member's name or email address > Share
Honestly, I didn't find Microsoft official article to explain this behavior. Very sorry about that.
I believe this issue needs to be investigated and fixed from the backend. So I sincerely recommend that your Microsoft 365 admin, can follow the steps in this article and create a support ticket to contact online support. The support team there has more permission than us. This would be the most efficient way to handle this case for you.
Note: To create a support ticket, you need to have Microsoft 365 admin privileges. Reference: How do I find my Microsoft 365 admin? - Microsoft Support
Please accept my apologies for redirecting you to the related team support, as the moderators in this community have limited resources to verify the backend information and to get the quick and better assistance, we requested for this.
Your understanding and patience will be greatly appreciated. I hope you're staying safe and well!
Sincerely,
Tina | Microsoft Community Moderator