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.