I have the approval block working in several other flows, which is why I'm thinking it's related to the For Each block, not the approval block itself. For this issue, there's 2 different variations that are both having the same trouble.
In the first scenario, I'm reading the members of an O365 group appending the emails to a string field to build the "assigned to" block of the approval:


As I mentioned, the approval is sent to all of the group members properly, but it isn't registering as a response being submitted.
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In the 2nd scenario, I'm setting a multi-person "assigned to claims" field (3 users - see below) on the SharePoint record, then referencing that field on the "Assigned to" field in the approval block.


Again, all 3 "assigned to" users are receiving the approval and are able to respond, but the flow is not continuing after the first user replies.
The flow is too long to provide a screenshot of the whole thing, so let me know if there are other specific pieces that you'd like to see.