Two attackers flanking get advantage due to the target being distracted and overwhelmed, right?\
Now a third attacker comes in, presumably leading to the victim being even more distracted/overwhelmed, yet that third attacker gets no advantage, either figurative or literal.
That's not it. Flanking specifically represents someone trying to get
behind you. Unfortunately, while the game can tell us positions, it doesn't represent facing at all, so there's no way to actually know whether someone is actually in front of you or behind you.
When there are two enemies directly opposite each other,
either one is on your front half and the other is behind you,
or they are both in your peripheral vision. The flanking bonus assumes the latter case. (There was actually an option, later in 3.5, to choose which way you face; the enemy in front of you gets no bonus to hit, but the one behind you gets +4 and you lose your Dexterity bonus against them.)
But then, if you have an enemy to your left and right because you're splitting the difference and not allowing either directly behind you, then a third enemy who joins in
might be in front of you. (And assuming that you can pivot freely, that's exactly what you would do.) Thus, the new enemy doesn't get a bonus at all.