Yaarel
🇮🇱 🇺🇦 He-Mage
Yes skills demonstrate non-magical competence, but there's nothing about "fighting" involved with most skills. I don't think they deserve extra skills.
Fighters have lives outside of combat training, and deserve more skills to be competent in the other two pillars of the game, Exploration and Social. Two skills is insufficient.
Also, many skills are directly relevant to a Fighter.
Arcana. In a world of magic, a Fighter must understand how it works and how to defend against it.
Persuasion. Combat leaders of any rank require leadership skills. The flavor of the Fighter is often a popular hero. Charisma skills matter. Insight is there, so social skills are a thing, already acknowledged to some degree.
Investigation. Understanding the weakness of an opponent, including an hostile camp, hostile culture, etcetera.
Medicine. How can a Fighter not understand first aid?
Sleight of Hand. Some Fighters are big in weapon tricks. Some are skilled at penetrating hostile territory, including unlocking doors, avoiding traps.
Stealth. How does a Fighter not learn from missions that require stealth? Every army in the world wears camouflage for a reason.
More than needing competence outside of combat in the sense of one or two more skills, the range of skills available need to expand somewhat.
If an entire culture is mainly martial, and teaches people to wield martial weapons like a staff, the staff is still martial.I disagree with this. A longsword is martial. A staff, while a weapon, is a basic weapon that any old person can learn to use. Wizards, being smart people, know that they can be in a situation where magic won't work and would learn some basic weapons to get by. It's not as if they even begin to remotely fight as well with their dagger or staff as a Fighter does with his weapons.
Alternatively, the Wizard has been studying arcane cantrips.
It seems fine to me, if a Background includes martial training. And besides, I already noted, the arcane Wizard should be able to swap out an arcane cantrip for a choice of martial weapon, if the player wants. That choice can be a staff. And use the staff magically as a focus. That choice can also be a sword.