Okay, let's play that game then.
Fighters learn the basics of all combat styles. Every fighter can swing an axe, fence with a rapier, wield a longsword or a greatsword, use a bow, and even trap foes in a net with some degree of skill. Likewise, a fighter is adept with shields and every form of armor. Beyond that basic degree of familiarity, each fighter specializes in a certain style of combat. Some concentrate on archery, some on fighting with two weapons at once, and some on augmenting their martial skills with magic. This combination of broad general ability and extensive specialization makes fighters superior combatants on battlefields and in dungeons alike.
Proposition 1: Fighters learn the basics of all combat styles.
Proposition 2: This means that they know how to use a wide variety of arms and armour.
Proposition 3: Fighters choose one skill out of the many they know to specialize in.
Proposition 4: Fighters are superior combatants because of this blend of breadth and depth.
Proposition 3, the specialization of a Fighter, does not invalidate Proposition 2, their wide range of skill. A Fighter may specialize in ranged weaponry, but that doesn't mean they're bad up close, and vice versa. And choosing a Fighting Style does not invalidate their global weapon and armour proficiencies.
As well, "Fighting Style" is never mentioned in the text. The closest mention of that we have is Proposition 3, but that statement also references the Eldritch Knight subclass, so it can't purely be talking about the Fighting Style class feature.
Finally, the Fighting Style class feature is additive to the Fighter's fighting ability. Specializing in one Fighting Style gives them bonuses to that style, but does not penalize them for using weaponry outside of their style. A Level 1 Fighter still has a wide breadth of martial knowledge and skill that in real life would take years to build up, even outside of their specialization.