It is a DBA responsibility to verify that database backups can be successfully restored but many DBAs (the majority ?) fail to do that.
This statement has not been true for many years now.
Every enterprise environment I know of has ALL backup creation, storage, retreival, and restores supported by employees outside of the Database or DBA team.
In Enterprise environments most DBA's wont have direct access to the application that creates the backup files or the place where they are stored. They also will not be able to restore the files.
Data theft or loss of personal and sensitive information through Backup files has become so out of hand that this has become the standard.
Its still a DBA's responsibility to make sure the database can be restored to a state where the data is valid and current.
This is validation is usually a checklist that is turned in after the DBA has validated the database schema and data after the restore is completed.
Usually these restores are only done during a refresh from the Production servers to the UAT servers, or during as DR exercise. I like the way this article uses that perspective.
I have never been in a DR exercise where the SQL server was restored and the msdb or master database was restored from a recent backup.
I would recomend and add to this that in Todays environment it is every DBA's responsibility to have a plan for restoring the system databases to a state where they can support the databases you are responsible for. In other words, you need to be able to get your databases working no matter where the Backup or Systems team restores them.