I restarted and am about level 10, have defeated Mirmilnur and discovered Dragonborn-ness, have bought the Whiterun house and all its accoutrements... but it seems that Proventus Avenicci, the Steward of Whiterun, does not have the option for odd-jobs (bandit whacking, etc) in his dialog.

I already did one job recently, one that you actually start by talking to the barkeep in Riverwood but collect from Proventus... is there a cooldown or something for that?

Did you ask him for a job immediately after you picked up a reward from the last one?
I think some time has to pass before he gives a quest again, sometimes only walking in and out of the palace may help, I noticed that in Riften.

No I haven't asked for a job from him immediately afterwards... Oh dear, hope he isn't broked...