Frankly, I'm not up to the task of deciphering which timeline did what to who at this point. It seems fairly clear that, with or without prodding from a red dragon from the future, the dragonflights eventually noticed that the night elves were up to what eldritch sages call 'something hinky' at the Well of Eternity. Neltharion, the aspect of earth and one of the most respected of the aspects, proposed to his fellows that they create a powerful artifact and each flight imbue it with a portion of their own power.
He didn't mention that he'd gone insane from his long exposure to the maddening whispers of the Titan's old enemies, the Old Gods, from their prisons below the surface of the earth that was his responsibility. Oops. Bit of an oversight, that. So each dragonflight (even the red) save the black dragons imparted a portion of their power to this 'Dragon Soul' and when the dragonflights finally sought to oppose the demons of the Burning Legion, Neltharion showed his true madness. He turned the power of the Dragon Soul against his erstwhile allies and went so far as to destroy the great portion of the blue dragonflight (the very destruction that claimed the life of Sindragosa, Malygos' first consort, in fact) -- the devastation caused by the Dragon Soul was so great that even after escaping its influence, the dragons were effectively forced into hiding while the Legion ravaged Kalimdor. (The old gods that had poisoned Neltharion's mind wanted the Legion to succeed in hope that Sargeras' efforts to enter Azeroth physically could be hijacked to free them from their ancient prison.)
Eventually, the mortal races managed to steal the Dragon Soul, although not until after it was used by Queen Azshara and her servant Xavius (thanks to Illidan Stormrage, who had helped them acquire it) to help widen the portal Sargeras sought to use. In the end, the Well of Eternity itself was destroyed to prevent the Legion from claiming Azeroth, and the remaining dragons sealed the Dragon Soul away behind magics that would prevent Deathwing (as Neltharion was called after his betrayal) from ever using it again. They also started calling it the Demon Soul, perhaps again due to the interference of a time traveling red dragon who really should have known better.
After the Sundering the red dragonflight was involved in the creation of the world tree Nordrassil in order to prevent a new Well of Eternity from bringing the very same Burning Legion they'd just gotten rid of by blowing up the first one and splitting their one continent into three pieces. The various dragonflights then spent a lot of time hunting down and killing black dragons, including the reds, but this war among dragonkind was one of horrible attrition and no flight made it through without heavy losses. This basically ended any hope any of the dragonflights would ever have of being able to truly control the destiny of Azeroth. Their ranks diminished, the red dragonflight's former policy of laissez-faire became the only policy they could adopt. Whereas once they allowed the world to more or less organize itself, they now lacked the numbers to step in unless they had no other choice.
The War of the Shifting Sands shows, in fact, the level a threat would have to reach before the reds would intervene. The Qiriaji, descendants of the Aqir and lead by C'thun itself, threatened to destroy all of Kalimdor. Rallied by the bronze dragonflight the reds and greens stepped in to aid the Night Elves against the old god and its minions. Even the blues, devastated as they were, sent aid in the form of the dragon Arygos. In the end the reds lost Caelestrasz, son of Alexstrasza (He's also presumably the son of either her elder consort Tyrannastrasz or Korialstrasz, her current prime consort) when the barrier that sealed Ahn'Qiraj was raised.
Things settled down for a thousand years or so after that. Of course, periods of relative peace couldn't last while the mortal races got up to their usual hijinks. Before the reds knew it, there was a dark portal and little green things were coming through it and there were mortals warring everywhere and it was all relatively unimportant to them until once again the Demon Soul popped up, thanks to Deathwing.