The DM determines what knowledge your character has, and then uses a check to measure your ability to recall something from that knowledge base. A fighter with no religious background, and no proficiency with Religion as a skill, should not be granted the opportunity to randomly recall nuggets of information. -- This is where the whole "randomness produces wonky outcomes"-thing arises.
I do agree with this to an extent. Sometimes I'll call for certain checks for only those who are proficient, or single out the character that took the lead. But if it feels wonky to you, find an explanation that works.
So even if the Barbarian manages to ace his Arcana check when the Wizard flubs it, as the DM don't just tell him how he knows that. This Outlander Barbarian somehow came across a rare piece of Arcane knowledge. How? Let the player quickly make up something interesting and keep it in mind. It's now part of his background, permanently. Whether it ever comes up again or affects anything is up to the DM and player, but some piece of the Barbarian's past is now set.
But I like to keep backgrounds pretty general and let my players fill in specifics over time, so YMMV.