You are conflating "skilled" with "thematically appropriate."
No I am not. Your idea of thematically appropriate is different than mine. I think it is appropriate for Gandalf to use his sword all the time in LOTR. I think a Wizard sailor is appropriate and will go a step further and point out that they have been in a published campaign since the 80s (Saltmarsh) and in the D&D novels.
The player, and ONLY the player, decides what is thematically appropriate for his character within the bounds of the rules.
Sure you can give a fighter a 20 intelligence, proficiency and expertise at arcana, but that will never make it appropriate thematically for the Champion or Battlemaster that does it.
Why not? If he has a 20 intelligence and has trained in the Arcane he is by definition an expert in it. The Wizard who took Warcaster, a Constitution feat and history and investigation needs to get over it and stay out of the Fighters way when talking about things Arcane or at least recognize she is second class at it as it comes to the party and defer to the Fighter on it.
Where in the PHB does it say that players can just swap out class or background skills like they want to?
Page 125, under Proficiencies and as I said it is not listed as optional. It also applies to tools, so if your Rogue has a criminal background she can take proficiency in say Cooks Utensils instead of Thieves Tools.
No dude. You get history and arcana as a sage because that's where you got those skills. You do not get "skulking" and "explaining yourself" from being a sage. You get it from being an urchin or something.
You get it from the rules and the player decides the thematics around it. This is the rules of the game as written and IMO as intended.