I don't think Varys knew the hatching of dragons was even possible until dany came along.. Still he does hate magic, and as Targeryen used to be a family actively involved in magic in the past, maybe his grudge comes from the fact that Dany's ancestors did something to his ansestors. Game of Thrones is like a big history show based on even bigger "history" books, things that happened in the past come back bitting everyones butt. Oberyn and his sister's revenge is an example, up until season 4 I didn't even know the people from Dorne hated Robert or the Lannisters. The past is important, and we have had glimpses of Varys's past for a reason.
But you make a good point, the eastern cities may be involved in some kind of conspiracy, I find no other reason of why a very rich man from the east such as Illyrio could actually care about the Iron Throne. And I still think a Dothraki army is NOT a good gift for Dany, he could've hired a sellsword company for them such as Stannis is doing. Illyrio is a man that just "gives away" three dragon eggs, he certainly has the money to help Dany and her brother with ships and an army, and I mean a true army fit to fight the west, where they use armour, and chainmail and big fortresses... no matter how many Dothraki attack the west they will never be able to conquer it just with their horses.. You might say they would have raided every village and gained enough gold for ships, but they are not sailors, they know nothing about marine strategy of even war strategy.. they just use strenght and raid small villages that have no way of defending themselves..