Sql-server – Include stored procedure text output in the maintenance plan log

The SQL Server Agent account has both share and local security permissions right?

I would also ensure that you have given the SQL Server Agent appropriate network permissions -- it really sounds like a permission issue.

You can get the backup date for each database backed up after the plan started from

--INSERT INTO YourBackupLogTable
--     (DatabaseName, LastBackupStartDate, LastBakupFinishDate, LastBackupDurationSecs)
SELECT db.name AS DatabaseName
    , MAX(lb.backup_start_date) AS LastBackupStartDate
    , MAX(lb.backup_finish_date) AS LastBakupFinishDate
    , DATEDIFF(S, MAX(lb.backup_start_date), MAX(lb.backup_finish_date) ) AS LastBackupDurationSecs
FROM sys.databases AS db
JOIN msdb.dbo.backupset lb ON lb.database_name = db.name
WHERE lb.backup_start_date >= (SELECT MAX(l.start_time) AS Start 
                                 FROM msdb.dbo.sysmaintplan_plans p
                           INNER JOIN msdb.dbo.sysmaintplan_log l ON p.id = l.plan_id
                                WHERE name = 'Your backup plan')
GROUP BY db.name