Symantec Backup Exec runing DBCC CheckDB and Backup on my SQL server
I'm not sure if Symantec still owns it, but Backup Exec is Netbackup's little brother. I don't remember the feature differences but Netbackup is more for Enterprise environments and Backup exec for smaller ones.

Are you sure it's Backup Exec that's running DBCC? I've used Netbackup for SQL for many years and it's always done checksum checking during backup. I've never seen it run a DBCC check directly from the Netbackup server or via the client.

Any third party SQL backup software rides on top of the MS written SQL backup. What you're paying for is much better compression, AlwaysOn support, centralized job scheduling and backup image storage. SQL backup has a checksum option and most of the third party products support it as well. But the only time i've seen something not-SQL run dbcc or alter index on a database is Sharepoint doing it.

I've used both and prefer the third party products to native SQL backup any day.