If an existing character, say a Fighter, decides to take a level of wizard with no prior mention of learning magic or interacting with wizards, and in a wilderness with no opportunity to find another wizard, would you let them go ahead?
The point is that players are supposed to apply common sense here too, right? I could see how you could have a 16 year old with that background if they'd been raised a criminal from an early age, or something like that. Or as has been suggested you could reframe it to work along the lines of a...
It says 'you are an experienced criminal with a history of breaking the law. You have spent a lot of time among other criminals and still have contacts within the criminal underworld'.
That's the funny thing isn't it. What makes the background traits so good is that they help set up all kinds of cool situations like this. 'No they dismiss you out of hand' is not only a loss of player agency it's a loss of colourful things to play out.
In fact we know the answer to this already. The player chose the background that says 'you have access to a network of criminal contacts' rather than something like an orphan or a craftsman etc.
There are things within combat that automatically succeed though. You can automatically move your base rate to get into close quarters with someone, for example (barring some sort of legendary interrupt action I guess). You can draw your sword automatically. You can use action surge...