Monitor source SQL code of inserts into table
You haven't included version or edition info, so if you're running SQL Server 2016 SP1 or higher (or Enterprise Edition for earlier versions), try using SQL Server Audit to trace this information.

You can set up a Database-level Audit to capture INSERT action on that table, then query the audit files to identify the calling application and statements.

SQL Audit: https://docs.microsoft.com/en-us/sql/relational-databases/security/auditing/sql-server-audit-database-engine?view=sql-server-2017

Audit Action Groups: https://docs.microsoft.com/en-us/sql/relational-databases/security/auditing/sql-server-audit-action-groups-and-actions?view=sql-server-2017

Also, Profiler and Extended Events should be able to capture this information for you. Which trace events are you capturing?

If you use the template TSQL_Replay trace and filter it appropriately, you should be able to see the calling code, including the explicit values passed to the stored proc or INSERT statement.