You can choose to play a smith's apprentice who's never held a weapon before in his life, but he still has the skills of a well trained warrior because you chose Fighter as his class. You can choose to play your halfling Bard who's only just leaving home, but she's still a well trained musician (proficient in three instruments) and already trained in a very specific form of magic. This isn't "background and story trumping hard rules", this is the opposite, this is game mechanics trumping background and story.
What I mean is, if your background and story doesn't agree with what the rules say you should have, then generally speaking, the philosophy of 5e design (as it's discussed in the handbook and the dmg for setting up the world, running the game and characterisation) is to allow the background and story to trump the rules. They recommend DMs and players lean into their personal story more heavily and agree on alterations that suit; my bard, for example, doesn't have full control over her magical capabilities, or a good understanding of them yet; I've got the mechanics in the background, but the in-game reality is that I don't have on-demand access to some of that stuff yet, even though by the written rules I should, and sometimes it happens by accident in stresfull situations.
