The whole Illyrio/Varys conspiracy stinks of a giant plot hole to me that GRRM will probably largely ignore. Specifically, I don't think we'll ever get a clear answer as to what exactly they intended Viserys/Dany to do with the Dothraki or how it relates with the Aegon/Golden Company invasion.
I think the end game was always Aegon.
Both Varys and Illirio wanted to set up a stable kingdom somewhere under a just ruler because of some ideological dream they both share combined with the kind of megalomania you only get from having risen as far as you can go. They couldn't have set anything like that up in the Free Cities because those places were too disjointed, both culturally and politically and a ruler's hard power only extends as far the city limits, and even then, rival power centres (priesthoods, guilds etc) divide your authority further still.
Contrast this with Westeros, where, despite the divisions, rulers have some measure of control over their vast lands. The Targaryens ruled a kingdom big enough to engulf not only all the territory of the Free Cities and Disputed Lands but Andalos to boot. One king, one set of laws, one language and one culture. If you were aiming to maximise your impact on history, Westeros is where you'd do it.
They hear that the king of Westeros is insane but that his son is well-liked and they smell an opportunity to make their dream come true. So they came up with a plan.
1) Get in good with the Royal family
2) Fill the heir's head with dreams of destiny, just rule and utopia, making yourself his most invaluable advisor, friend and greatest champion. At the same time, make the king himself the most hated man in the realm.
3) Engineer the king's death
4) Profit!
The only problem with the plan was that it almost worked
too well. Rhaegar let the prophecies get to his head, yes, but he also convinced himself that getting some Northern poontang was the only way to fulfil his destiny. Had he never done that,Robert wouldn't have hated him and his rebellion would never have been against the Targaryens in general, but Aerys in particular and in fact Robert and the rest of the high lords might have accepted
Rhaegar on the throne instead.
Basically, Lyanna ruined everything.
Trying to pick up the pieces of the plan, Varys and Illirio set their sights on the next generation of Targaryens. They know that Baratheon hegemony is unstable, so baby Aegon is their horse. The new plan:
1) Make Aegon into the second coming of Rhaegar
2) Bide time until he comes of age
3) Destabilise the realm by killing Robert. Cause chaos.
4) Raise an army to invade with Aegon at the head
5) Invade and take over. Join the new regime as their most trusted advisors.
6) Profit!
Where Dany and Viserys came in was in step 4. The sub-plan was:
4.A) Make a deal with Khal Drogo to invade Westeros in exchange for a Targaryen bride
4.B) Send some assassin out to kill Dany and any of her children. Make sure the assassin is Westerosi and that it's absolutely clear that King Robert is behind it.
4.C) Vomiting with rage, Khal Drogo will drop everything to rush to Pentos, to Illirio and Viserys, 40,000 Dothraki screamers at his back.
4.D) He will arrive to find Viserys mysteriously missing and/or dead. That doesn't matter though, since Dany's long-lost nephew is here. AND he conveniently has ships.
4.E) Invasion happens. Khal Drogo gets his revenge on Robert (and a lot of booty). Having no interest in actually ruling Westeros and having no heir to give it to, Khal Drogo goes home with lots of treasure, his nephew on the throne.
Varys and Illirio thought they could pull it off. They, after all, just came off the big leagues of political intrigue in the Free Cities and probably thought that their Westerosi equivalents were novices by comparison.
Thing is, they can't plan for or control everything. The dragons, for one, or Viserys going with Drogo to the Dothraki sea, Robert's impetuousness in wanting Dany dead, or Khal Drogo dying from an infected nipple scratch and his army fracturing / dissipating thereafter.
Chances are, after Drogo died, they just wrote off that part of the plan. Dany starts taking over Slaver cities and suddenly she becomes useful again. Maybe they can co-opt those dragons by marrying her off to Aegon. It would certainly lend him some extra legitimacy, which can't hurt.
Point is, even if that doesn't work, Dany is not a critical part of the plan. The main thing is to destabilise Westeros and get it ready for invasion by Aegon and the Golden Company.