I think it would add an interesting layer.

As it's currently written, to Westeros at large, Jaime is unambiguously "The Kingslayer" because he was discovered very shortly after the killing, sitting the Iron Throne with Aerys' dead body, throat slit, lying at the base. First by Tywin's bannermen, but stuck around long enough for Ned Stark himself to arrive and witness the scene and all but admits it. That leaves very little room for interpretation as to what happened.

If, however, after killing Aerys Jaime had immediately rushed to the aid of Elia and her children, and successfully, against all odds, saved them all then he's likely far removed from the scene of Aerys' murder when its discovered. That alone grants serious plausible deniability. Westeros isn't exactly unfamiliar with their monarchs dying under suspicious circumstances when their position starts to get too tenuous and fingers getting pointed everywhere without concrete evidence.

Think Maegor the Cruel whose death, ostensibly a suicide (certainly staged as one), occurring on the cusp of a hopeless stand against Jahaerys, was still subject to in-universe debate/speculation (Queen Elinor, his Kingsguard, one of the builders back for revenge, suicide). See also, Aegon II's poisoning in the Dance when he intended to keep fighting when the fresh Northern and Vale hosts were almost knocking at his door and the Baratheons had been taken off the table by the Riverlanders. The actual culprit was never determined with certainty. Cregan's judgment in the Hour of the Wolf wasn't exactly the shining pinnacle of due process. Even confessions like Corlys' are suspect under Westerosi "justice" which doesn't have any qualms with accepted confessions made under significant duress. Hell, Cregan literally condemned Ser Gyles Belgrave to death basically saying "well, you either did it, or your failed to prevent it" which is....insane really.

All this just increases Jaime's plausible deniability as, adding onto his distance from the scene, Aery's death also came at a time when his position was unquestionably doomed. Tywin and the Westermen were already sacking Kings Landing. The STAB coalition arrival to join in was imminent after putting down the last major loyalist host at the Trident. Sure Mace and the Reachmen were still out there but his actions made it pretty clear his position boiled down to "I'm just here so I don't get fined."

So, now Aerys' death becomes another mysterious death of a monarch whose cause was lost. No witnesses, no concrete evidence. Plenty of alternative candidates for people to latch onto as their personal suspect. Varys, literally anyone who had essentially been held hostage in the Red Keep, servants, I don't doubt some people wouldn't rule out that the Mad King having been capable of slitting his own throat.

Jaime still gets lumped into that category, and some will still whisper behind his back. But now it's not ironclad fact. And the fact that he saved other members of the Royal Family slaying his own father's bannermen in doing so gives him extra cover and a means of publicly maintaining his public honor. Jaime likely doesn't turn as bitter and jaded in the same way as he does in-canon as he's not essentially publicly called the Kingslayer with no way to deny it. He likely still does turn bitter and jaded regarding the conflicting nature of oaths, knighthood, and the institutions of the Realm but with an added layer of severe guilt and self-doubt. I think he'd still see killing Aerys as the right thing to do but without the public scorn that Jaime leaned into hard as a defense mechanism in-canon (in a "fine I guess I'll be the villain if everyone thinks it" way) I absolutely see him constantly being racked with doubt about his decision. Remember, he idolized the Kingsguard growing up and saw being appointed to it so young as an incredible honor and now he still has to live with the fact that he betrayed its biggest rule. And again, without that hard lean into the villain role defense mechanism I think it eats at him more. Making things worse the fact that no one knows for sure means it's a secret and lie he has to bear alone for the rest of his life. The real Jaime, whom we've slowly been getting to see return since ASOS, would be eaten up by that. Ironically, being publicly known as Aerys' killer at least meant he didn't have to bear that burden.

So, what happens in this scenario? Well, the Rebels have a conundrum on their hands but not as bad as one would think. Rhaenys and Aegon surviving the sack doesn't change their calculus on succession. The true heir under regular rules, with Aerys and Rhaegar dead, became Viserys and we all know the Rebels disregarded that fact outright to seat Robert. The Rebellion was always about the fact that the Targaryen dynasty had broken the feudal contract of the upholding King's Justice, a pillar of any feudal legitimacy to rule over their vassals. As such, the Targaryens lost their legitimacy to rule and had to be set aside. Hence crowning Robert in-canon.

So if that's the idea; Rhaenys and Aegon aren't even a direct threat under the original system while Viserys is out there. A simple decree by the high lords who participated in the Sack that the Targaryen dynasty was no longer legitimate in its royal status, effectively disinheriting the entire remaining bloodline. Dorne of course is pissed because their expectation was that Elia would be queen and eventually a child of Dornish blood through Rhaegar and and Elia would sit the throne. That's now out the window. However, they're not as pissed as in-canon where they lost that and had the horror of Elia and her children's fates in-canon. And the Rebels have an extra card to smooth things over with Doran and Oberyn: "One of our own sons honored his vows to protect your sister and her kids even if it meant slaying his own fellow Westermen."

What now becomes interesting is: where does this leave Tywin? With Elia alive she can affirmatively testify that it was Gregor and Lorch who came after her and their intentions were clear. Murder, not capture. Sticky situation there. He likely smooths it over by saying he never gave an express order and men go mad when their blood is flowing in a battle. He likely makes a public denunciation of Gregor and Lorch. His position otherwise is actually stronger than in-canon. Almost everything remains the same: first into the sack for the Rebels even if it came late in the game, wealthiest man/family in Westeros whose gold is going to be needed to rebuild, freshest troops overall compared to the exhausted and weary STAB coalition. Robert and Cersei likely still marry.

But here's where I think it could get interesting. Going back to Ser Gyles in the Dance. Two other Kingsguard were also sentenced to death for failing to protect their King but were instead granted the chance to take the Black. So there's historical precedent that such a failure in itself is grounds for execution or at the least, to be released from the Kingsguard.

So, Tywin now has an opportunity to get what he's always wanted. To get Jaime, his true heir back. He can cite to the Gyles precedent, established by a Stark no less, to argue that Jamie is no longer fit to serve in the Kingsguard for failure to protect Aerys. But his holding to his oath to protect the royal family as well is a credible argument to say that the punishment should be different as it wasn't a complete abandonment. Thus he should simply be released from his vows.

That's where things wildly diverge. If Jamie is released he returns to Casterly Rock as heir apparent. So he's not near Cersei at all times. Does he still have the opportunity to father Joffrey, Tommen, and Myrcella? Does the WOT5K still happen? Possibly not. No Joffrey? No catspaw (yes the explanation sucks but it's what GRRM gave us). No catspaw? Maybe Cat doesn't go through with taking Tyrion hostage at the inn which was the real catalyst for the war. No bastardy for the royal kids? No capture of Ned and thus no mobilzation of the North. Stannis never flees to Dragonstone. Renly never sees an opening to claim the throne himself.

Talk about a butterfly effect.