Your issues seem to be more political than anything else. If there is a time when you have a drive failure, you'll be thankful for the off-site backups.
With regard to maintenance, Transaction Log backups are always a good idea, providing the database is in Full or Bulk Logged Recover Model. If not, a Full backup weekly with Differentials daily may be adequate.
I say "adequate" because I don't know how volatile the data in your Database is. The more changes, the more frequently it should be backed up because with lots of changes usually comes a heavy reliance on that data contained.
As for me, I'm always thinking rainy day scenario when it comes to recovery. By planning for the worst case, your almost always covered for the little hiccups.
I take a full backup weekly, Differentials every day, and Transaction Logs every 15 minutes. All of my backups are written to local disk for the sake of expediency, but I copy those files to another Server as soon as the job has completed and hold them for 2 weeks (I have the space to do this so why not). I also write those files to a tape archive system and hold them in a 45 day rotation via an off-site storage company.
I have recovered from some nasty disasters. The worst is losing all of the Servers in the basement Data Center in a flood. Every Server was gone, but we were able to recover to within 1 hour of failure because of the number of backups and where we copy the files.
Food for thought.