There's not much to go on regarding the Society's actual plans in WoR and its supplement.
There may have been no single, unified, enforced set of Society goals and playbook. WoR (p. 152) states that "only a few among the top select members knew the objectives and directions the Society was moving in." If there was little top-down direction, every Society cell may have had its own plan. Given some of the Society's goofier moves (like the voice on Strana Mechty), that may have been the case. It certainly leaves things more open for GMs, which may have been the intent.
That said, if the Society's top leadership was able to agree on and enforce a master plan, it probably looked something like what happened to the Coyotes. Elliott McKibben, a member of the scientist caste who had tested out of the warrior caste, was able to retest, rejoin the warrior caste, and rise to the Coyote Khanship, apparently through Society subterfuge and threats.
If the Society had been able to subvert the toumans in a few additional Clans besides the Coyotes (augmented more of the Society's own specially equipped forces and tricks like HPG and warship virii), the Society would have had a reasonable shot at taking over or separating from the Homeworlds by force. But they obviously did not have enough time to do so before the Bloody ilKhan's Reavings started.
Without enough time to subvert more of the warrior caste, I think the only other option open to the Society was more widespread use of their DNA-targeted virii. The Society could have cut down many more bloodhouses and even whole phenotypes if their genetic weapons had been more widely deployed, effectively wiping out the warrior caste. But the Society was probably loathe to do so because they literally had centuries invested in the development of those bloodhouses and phenotypes. I imagine the Society only targeted individuals and bloodnames that they considered to be failures. The Society's geneticists wanted to bring their best creations (the best trueborn warriors) under their control, not destroy them. (Plus, wiping out the warrior caste with biological weapons would have made for a boring storyline and gaming scenarios.)
It's also worth noting that many Society members were no doubt former warriors themselves and vulnerable to their own bloodname and phenotype virii, making them even more loathe to release those biological weapons into the warrior population less they be hoist by their own petards and die horribly.
Although the Society's battlefield technology and tactics are very interesting and flavorful -- iATMs and improved heavy lasers, ultra-heavy and modified protomechs, high-tech and highly efficient omnimechs, warship virii, etc. -- I don't think those tools alone could have won the day unless a lot more of the warrior caste had been subverted like the Coyotes or had been wiped out by the Society's DNA-targeted virii.
I also don't think that uprisings (or lack thereof) by other castes should matter. Logically, the BattleTech rule of sheeple applies to the Homeworlds just as much as it does in the Inner Sphere. Whether he's a megacorporation employee, a backworld farmer, or a member of the laborer caste, the average Joe just doesn't care who rules his planet. The vast Spheroid populations in the Clan occupation zones never rose up en masse to take out the cluster or two that garrisons each of their planets after dozens of years of Clan rule. Had the Society been more successful, I'd say it's equally unlikely that freebirths and lower castemen in the Homeworlds will have risen up against the Society. In fact, given the rate of trials of possession, some, maybe most, Clan enclaves are accustomed to being traded to a new team every so many years.
Lastly, the one battlefield potential untapped by the Society or the Clans in general is sheer numbers from production. If you only care about minimal quality, iron womb technology has always had the potential for turning out massive numbers of warriors. If the Society had paired that with mass production of low-cost but effective weapons (protomechs, small vehicles, maybe even battle armor), they might have had a chance even without subverting more of the warrior caste or wiping it out with virii. If you can send ten green or regular protos or small vehicles up against every veteran or elite omnimech on a planet, you'll win the planet more often than not. Maybe the Society never had the resource and facilities to achieve those numbers, but things might have been very different if they had.
My 2 Kerenskies... FWIW...