Chaos wasn't their method though.

The early story was a struggle between Varys and Petyr. Petyr was trying to destabilise the realm as quickly as possible wheras Varys was trying to maintain the status quo to solidify his own position until Khal Drogo was ready. Their methods are antithetical and one of the great dualities of the series. That's part of the reason why his decision to help Tyrion escape is so powerful, it's a statement that he's ready and his plan is finally in motion, albeit not as concrete as the original plan.

Varys benefits from chaos only once Tywin is dead and the major powers are militarily spent. Tywin's presence is the glue that holds the realm together but Tywin isn't a guy you can just shoot with a crossbow in the middle of the night when you're playing the long con, you need to maintain motivational distance from other power players, deniability from suspicious deaths and an alternate narrative should shit hit the fan. Varys had all of this, he presented his motives as 'for the realm', a vague statement that most would believe to mean that he served his King unequivocally, reinforced by his apparent nonchalance at the prospect of assassinating Daenerys. Maintaining the loyal image is more important than staying on track with the master-plan. By maintaining the image he has the freedom from suspicion to continually alter the plan unopposed, if he had raised concerns about assassinating Daenerys, it would have raised alarm bells for other major players and restricted his schematic freedom.

I've always had a hard time understanding why so many people are puzzled by his decision to support the assassination, or at least present himself to support it. It's entry level politics, not to mention there's no evidence that the actual assassin was of his making.

Sparking a war had a decent chance of weakening his own position, you could argue Petyr's War of Five Kings did weaken Varys' position. The opportune time for Robert/Tywin's deaths would have been immediately after the Dothraki landed, at that point Varys has maneuverability to take more Petyr-esque risks, at that point his image is of little concern as his griffins had already come to roost, win or die. Ideally he'd remain in his position until the actual storm of King's Landing but it's not a dealbreaker.

Ned couldn't help put a Targaryen on the throne, correct. That's why Varys immediately dismisses it after Illyrio's suggestion with "this hand is not the other". Illyrio has little knowledge of the intricacies of Westerosi politics.