Agreed with the sentiment about dragons sorta ruining the gameplay. While pre-Aegon's landing would make for a more divided Westeros, and thus plausibly more fun, I don't see how we can balance the addition of the dragons. The fact that we only know a handful of characters in that era doesn't help much either.
Aegon failed in his invasion of Dorne, and could have been conceivably defeated if more than two kingdoms had tried to fight back together. Argilac would have been fine if he'd just stayed in his bloody castle like every other defeated lord of Storm's End didn't. And in theory, Aegon also had the option of going east and forming a new Valyrian Freehold, and the Conquest might never have happened at all. There's wiggle-room.
And not knowing characters is no obstacle.
Also, while dragons are nasty win-buttons in Westerosi warfare, they're not invincible. The Targaryens eventually ran out, and it's unlikely the Dornish killed
all of them. Aegon also had to break out all three on the Field of Fire, so there are potential military situations bad enough where one dragon just wouldn't be enough. Really, the dragons should be powerful enough that they can be
just a plausible equalizer between the tiny Targaryen army and the large armies of the Westerosi kingdoms, with an equivalent cost so that you can't just make thousands of them. Extremely powerful, with some nasty associated events for killing characters, but difficult to get and hold on to and nowhere near so powerful as to be undefeatable.