Dragonblooded have all sorts of incentives to aggregate into the biggest groups possible. Their magic is substantially stronger when they work in groups. Further, while they have long lifespans they do need to reproduce, and to make certain their kids Exalt they need to reproduce with someone who has as high a Breeding background as they can get. The further out into the Threshold you get, the thinner and thinner the bloodline the local mortals get, so a sworn brotherhood that went out and overthrew a nation would probably breed their bloodline out of existence within a generation (or two if the brotherhood had absurdly high Breeding).
Aside from that, the Realm and Lookshy are aggressively recruiting from Outcaste Dragonblooded they find in the Threshold. Thin bloods that Exalt and the Realm hears about are likely to get a recruitment pitch in the form of whatever is required, and the Realm has so much to offer a young Dragonblooded that most will take the coin or the razor.
Finally, if your sworn brotherhood did run out and conquer a nation, what is likely to happen is someone from your Great House shows up within a decade and congratulates you on your youthful ambition, hands you a mandate from the Deliberative with the Empress’s seal on it naming you satrap, and then asks when you start intending to pay your taxes. Because this is a nice city you have here and it would be a shame if some Legions happened to it.