Automated regular tests of your backups
Provided that you have alerts set up on the jobs, Log Shipping is a feature that is available on all editions, doesn't require a whole lot of effort to set up, and I believe fulfills all of your requirements.

I would avoid any wizards that you might stumble upon in Management Studio. Essentially you need three jobs for the following tasks:


Initialize the secondaries (restore the latest full backup with NORECOVERY)
Take full and log backups
Restore log backups with STANDBY, NORECOVERY


Essentially, testing the restores of your full and database backups will be verifying them (and you can add other tasks to this process, such as bringing a restored database into recovery and checking data, running CHECKDB, etc)

I go into great detail about how I have implemented log shipping (different purpose, but generally the same approach):


Readable Secondaries on a Budget