True, but both Ralof and Hadvar clearly mention Cyrodiil. Something like "Surely, people in Cyrodiil have heard about the war in Skyrim!" and "I guess people in Cyrodiil have other things to worry about...".

How is that bad writing?

Not every Nord, Altmer, Breton, Orc, Dunmer, Bosmer, Khajiit, Redguard, Imperial, or Argonian is born in the province that their race calls home.

I assume that there are Imperials who were born in Skyrim as well so . . .

So please tell me how the Dragonborn being from Cyrodil is bad writing?

Bethesda never really explicitly states where the heroes are from except for arena if I am not mistaken but even then the hero in that game may not have been born in the province you start in.

The reason they usually do not state where the hero is born is the role playing potential.

The hero could have been born in Morrowind (an elven Dovahkiin could very likely have been born just after the oblivion crisis ended and their family could have moved to cyrodil at some point) if they were a Dunmer and still have lived in cyrodil at some point after their birth.

Gossip and rumors run everywhere. And some of the Imperial Legion was shipped out to Skyrim.

Unfortunately for her, my main character didn't pay attention to gossip and rumors, and wanted to go home.

I don't remember that dialogue, but in any event they could be wrong, just like Hadvar assuming you're from wherever depending on your race.

Anyway as people have pointed out, you don't have to be "from" Cyrodiil to pass through there.

I said that a long time ago.

The Jerall mountains is large place, not completely unbelievable that you happened to poke your head out of your cave to go hunting and end up walking into the middle of a civil-war skirmish during a blizzard.