SQL Agent Proxy - writing to logs is only allowed to jobs that are owned by sysadmin
I would work on educating your security team about jobs...they are server/instance level objects, not database level objects, so db_owner seems severely limited.

Ola recommends sysadmin for the job owners:


  Which permissions are needed for the SQL Server Maintenance Solution to work?
  
  
    If you are using SQL Server Agent, the jobs run under the SQL Server Agent service account that is a member of the sysadmin server role. If you are using a proxy account, I recommend that the account be a member of the sysadmin server role.
    
    If you are using another scheduler, I recommend that the scheduler run under an account that is a member of the sysadmin server role.
    
    If you need to have users execute the stored procedures ad hoc against specific databases, these permissions are needed:
    
    DatabaseBackup: sysadmin
    
    DatabaseIntegrityCheck: EXECUTE on dbo.DatabaseIntegrityCheck, VIEW DEFINITION on dbo.CommandExecute, VIEW DEFINITION on dbo.CommandLog, VIEW SERVER STATE, db_owner on all target databases
    
    IndexOptimize: EXECUTE on dbo.IndexOptimize, VIEW DEFINITION on dbo.CommandExecute, VIEW DEFINITION on dbo.CommandLog, VIEW SERVER STATE, db_owner on all target databases
  


Regarding the comment about the msdb filling up if you run against a table, when you install his jobs using his big script MaintenanceSolution.sql he installs a CommandLog Cleanup job that will clear out the tables used to log his job steps.