Why did Illyrio send Barristan to Dany instead of Aegon? Barry’s reputation is seen as useful for the ruler he would turn up with.

Well, keep in mind, I’m fairly sure Illyrio’s plan to get Dany to Pentos would have involved trying to get her together with Aegon and get her dragons under his control. 

Dany and Aegon showing up as independent rival monarchs was not part of the plan up to mid-ADWD, and if it hadn’t been for Tyrion you likely would have had Aegon showing up at Meereen as one of Dany’s suitors, like Quentyn but with sex appeal (sorry, @poorquentyn, but it’s true) and one of the best armies in Westeros at the moment she’s under siege.  

The problem is that Tyrion completely deconstructs Aegon’s Hero’s Journey, and so instead of the Hidden Prince Marrying the Princess and Training His Dragon, Aegon’s destiny is to become the Mummer’s Dragon who will take the Iron Throne to the cheers of the mob, only to be roasted alive when the Slayer of Lies comes to take what’s hers. 

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So the personnel shifts in Essos turn out to have been really crucial: sending Barristan means Dany doesn’t get assassinated on the docks of Qarth (which would have been a major setback, but would have cleared the way for Aegon ultimately), but because Jorah has his own plan which doesn’t involve Illyrio, Dany doesn’t go to Pentos, so that plan is scotched. The problem here is that Illyrio is trying to work at arm’s length (which is good tradecraft, but there are…well, tradeoffs) across two continents, and unlike in cyvasse or chess, the pawns have wills of their own:

“Which plan?” said Tristan Rivers. “The fat man’s plan? The one that changes every time the moon turns? First Viserys Targaryen was to join us with fifty thousand Dothraki screamers at his back. Then the Beggar King was dead, and it was to be the sister, a pliable young child queen who was on her way to Pentos with three new-hatched dragons. Instead the girl turns up on Slaver’s Bay and leaves a string of burning cities in her wake, and the fat man decides we should meet her by Volantis. Now that plan is in ruins as well.”

And so do the best laid plans of cheesemongers gang oft alay…

Dany won’t burn Aegon because: she is not a pyromaniac, has no proof that Aegon is fake, Aegon has done nothing to piss her off (even killed/defeated her enemies for her), Aegon is a kinsman offering her an alliance (burning him is not exactly great PR), Aegon is a Targaryen with public support etc. I can see potential conflict but it goes against Danny to do this sort of thing

Totally disagree. 

Dany’s decisions about a lot of things are driven by prophecy, as we see with the “three treasons.” And in Aegon’s case, Dany has prophetic reasons to believe that Aegon is false – the mummer’s dragon – especially when she turns up and Varys the Mummer shows up by his side and realizes that Varys and Illyrio have been double-dealing with her. 

And in terms of pissing her off – he’ll have taken her throne. Yes, he’ll be offering her an alliance, but as one of his (multiple) queens. Dany is not about that any more. She’s the Mother of Dragons, the hero heralded by prophecy, and she is done making peace. 

Could you expand on why you think Arianne will marry Aegon? When reading Arianne II it seems that Elia Sand is the more likely to get involved with him, based on her defiant personality, impetuousness, and exploring her sexuality. Arianne is more consumed by maintaining her status as heir to Dorne and wondering if Quentyn’s return would change anything. And Connington would never allow it; Aegon’s hand needs to be kept open, so if it happens it’s probably in secret and done quickly.

Arianne wants to achieve her father’s ambitions of destroying the Lannisters – “my father entrusted this task to me” – Aegon has an army in the field and is marching on King’s Landing right now.

Arianne doesn’t like the idea of King Quentyn – “will I need to kneel to him” – marrying Aegon makes her Queen, just as Doran had intended with his marriage pact to Viserys, and cuts out Quentyn too. 

Connington wants to keep Aegon’s hand open for Dany, but Aegon might not agree. Aegon’s whole deal is that he’s aggressive and ambitious and isn’t willing to wait – here comes Arianne Martell who has armies now, not “one day.” He’ll overrule Connington, as he did with Storm’s End and with Rolly, and he’ll use Arianne’s spears to take King’s Landing, seemingly vindicated. 

And then Dany will show up. Aegon might offer her a tripartite marriage, but she’s not going to be interested in negotiating. 

Aegon’s Crowning

The army he currently has is not enough to take and hold King’s Landing. If he tries, there is still the force along the Mander with Garlan to consider to crush him against the walls of King’s Landing. He needs to gather a large enough force to be able to oppose any other challengers. It is going to take a while to convince everyone he is the real Aegon, and gather enough support. 
Nothing in the vision explicitly says it is King’s Landing, so that isn’t a certifiable fact at the moment. King’s Landing isn’t the only city in the realm, there is also Oldtown. The first Aegon was crowned in Oldtown, and Oldtown supported Aegon II. The High Septon could come to Aegon, and crown him in the Starry Sept, the original seat of the Faith, with Aegon following in the footsteps of his namesake. They are trying to portray Aegon in the image of Aegon I. One doesn’t need to sit the Iron Throne to be king. Aegon the Conqueror was called “king” before the Iron Throne was even forged. To be called “king” one simply needs to be crowned. 
I think in the second Dance, Oldtown will be Aegon’s base while Harrenhal will be Dany’s. 
Also, there is one inconsistency. You say Dany is the younger queen that will cast Cersei down yet Aegon will take King’s Landing from Cersei, or rather cast her down. In that case the younger queen would be Arianne not Dany. So if Dany is the younger queen to cast Cersei down, then that means Aegon won’t take King’s Landing, since by the time Dany arrives Cersei will have already been cast down. 

Aegon has 10,000 men. The Martells have 25,000. Many of the Stormlords have stayed out of the war. And there are friends of the Gold Company in the Tyrells’ ranks. That’s all he needs.

Garlan’s army is a thousand miles away and busy fighting the Ironborn. They’re not coming to help.

Aegon is not going to Oldtown – King’s Landing is the capitol, the Iron Throne is the symbol of a unified monarchy, the High Septon he needs to crown him is in King’s Landing, and it’s less than 500 miles away. He’s not going to go 1400 miles out of his way.

I think Cersei’s going to be awaiting trial at Aegon’s pleasure, thinking Arianne was the queen she didn’t see coming, when Dany arrives. It’s a classic prophecy fake-out, the same way that Tyrion isn’t the valonqar.