Sorry, quick update/clarification. After running some more tests, we found that this issue is not related to a specific collection in the DB. Even with balancing disabled on the apparent problem collection, the exact same issue arose, but with a different collection.
So, some issue here where the data migration process between shards is somehow causing the cluster to become unresponsive to queries and spool up a massive number of connections/threads while migrating chunks between shards.