The +1 may affect every attack but makes a difference with relatively few. Flying neutralizes all ground-based close combat. Advantage: flying.
Again, sort of. For one, in most indoor combat, flying is pretty much useless. And, let's be honest, a lot of combat does occur indoors. Plus, any character focused on melee has just pretty much neutralized himself by flying. Flying armor doesn't do that 3.5e two sword wielding ranger any favours forex. Even if that +1 only comes into account 5% of the time, considering the number of attacks, that's still going to be significant.
The same can be said of charm person, yet I think most people around here would say the charm is more powerful than +3 armor too. In this case, I'd still give the advantage to flying.
Really? You think a Charm Person is more powerful than +3 armor? Charm monster? Ok, I'd buy that. But Charm Person? It only affects a tiny slice of the creatures that you're going to face, has a saving throw and costs you an action to use. And, even if it succeeds, isn't guaranteed to do anything other than stop someone from attacking you - it might still attack your allies depending on your Cha check.
Charm Monster as an at-will effect? Ok, that's very powerful. But, that's due to the nature of save or die effects, more than the spell itself.
I notice that you picked a very different power this time. Swimming isn't in flying's class... Outside of a campaign that offers a lot of opportunity for undersea adventuring, that is. But how about armor that offers freedom of movement? A significant bonus to movement? Protection from fire? Then, +1 or +2 aren't a dominating strategy at all.
I like basic pluses to armor and weapons. They are simple. Their effects are relatively easy to control and understand.
Let's go through your examples. Freedom of movement is edition dependent. Outside of 3e, it's pretty much pointless as either very little actually can grab you (AD&D) or the effects of grab aren't terribly strong (4e). So, outside of 3e, I'd say +1 AC is a much bigger, certainly more often used, effect.
Significant bonus to movement. Ok, let's say it doubles your movement. Again, who cares? It's so situational that it isn't going to come into effect that often. Most encounters occur at distances under 100 feet. Being able to move double isn't going to have that much of an effect. Particularly in 3e where the costs of movement are so high.
Protection from fire? You're kidding right? Again, who cares? It's nice to have and all, but, it's going to come into effect FAR less often than that +1 armor.
The most significant effect you can have is any static plus. Doesn't matter what the plus is for, it's that plus that will have the broadest effect. It has to. Protection from Fire (for example) only matters when facing certain creatures, but, a +1 sword kicks in every single attack. We're talking about affecting literally hundreds of die rolls vs dozens (at most).
Sure, it's easy. But, the effects are so broad reaching that an entire edition had to base all its math around the presence of those pluses. Because, at the end of the day, it's not that +1 armor that's making the difference. It's the +1 armor, +2 weapon, +1 amulet of natural armour, etc, that every single character in the group has.
Between stat pluses and straight up equipment bonuses, a character is effectively adding levels onto herself. What's the difference between a 4th level fighter with +1 weapon, +1 armor and +1 shield and a 5th level fighter? The armor bonuses make up for the HP and he's attacking exactly the same. 2 skill points? That's about the entire difference.
Would adding Protection from Fire and a Sword that Casts Light really add an entire level to a fighter?