Take a better look at what Gang Up says:
"You are considered to be flanking an opponent if at least two of your allies are threatening that opponent, regardless of your actual positioning."
No one else is "providing the flank." It flat out considers you flanking if two other people are threatening the creature. It replaces the standard position requirement with two other people threatening. This completely replaces the normal rules, as the Normal section calls out.
In the next case, when the armed combatant makes the attack, both characters are considered flanking by the rules. They aren't immediately afterward, but by the rules two characters must flank for one to get the bonus. Both are flanking when that happens.
Nothing requires you to benefit from flanking to be flanking.
You're taking a concept the rules don't support and using it to explain why the rules work. You've said it before, you'll say it again, it still doesn't work.
When you attack, whoever is helping you flank is also flanking at that time. Two characters flank.
What you are trying to say, is that when only one person gets the benefit, the other is providing the flank rather than flanking. Where do the rules state this? They don't. Instead, the rules say that they aren't flanking, until they are briefly, then they aren't flanking again (until they are again, rinse and repeat).
There is no third, providing the flank situation, there is simply flanking or not flanking at any given time.
"When in doubt about whether two characters flank an opponent…"
Does this not specifically call out two characters flanking a target?
If only one character flanked, it would call out two characters. It would read similar to:
"When in doubt about whether a character flanks an opponent, trace an imaginary line between the the character's center and his ally's center. If the line passes through opposite borders of the opponent's space (including corners of those borders), then the opponent is flanked."
Then, you could argue that only one character flanks at a time, and that providing the flank is a real thing. The rules don't read that way though. They specifically make it clear that both characters just flank. There is no heightened priority to the attacker over the non-attacker.