If you want the biggest prize you must play the longest game...the one that has the highest stakes. Over the cousse of years, GTI would place their loyalists in key place throughout the EA. Given the ultimate size of the EA, it's number of devisions and the size GTI branch at the beginning.
It's like asking the UK to infiltrate an entire colonised star system. It could be done, but would take a long time. to get sufficient number of loyalists into position, so when to coup is pulled off, it only takes about a day or two.
The EA was also a totalitarian regime. The Icanus was the EA's untimate weapon - almost a planet killer. The symbol of their power, like the Death Star to the Galactic Empire in Star Wars. If the GTI waited till the EA was a it's height of power and then launched their coup, given the level of infiltration, the EA would fall to GTI quickly and GTI would look like they were coming in on the provebial "white horse" many of those agaist the EA would probably support the GTI