Spartans are the most versatile military faction in my experience. Since their banned civic is Wealth, which is only rarely preferrable over Knowledge, they can go for pretty much any victory they want.
You are correct that early aggression is the key for Spartans, but after they have conquered their first or two opponents there's nothing stopping them from transitioning into a domestic powerhouse.
I'm going to assume we're playing on the highest difficulty. Democratic Planned economy Spartans get +4 Growth for population boom. +1 Police from their faction bonus lets them neutralize 2 extra drones with Non-Lethal Methods. Assuming no wonders, and only basic Recreation Commons, this gives Spartans +6 drones neutralized, which incidentally is the population cap without a Hab Complex.
Our strategy then is to conquer one or two factions with our aggressive early game bonuses. Transition into Democratic Planned economy and population boom all of your conquered cities into 7 population. Once you're population capped, switch into Police State Green economics. With +3 Police your police units now neutralize 3 drones each. This lets you grow your cities to population 12 without any drones. Add Hologram Theaters and you're at 14 pop which is the cap without Habitation Dome.
At this point you can either keep going as is, invest into a military to conquer the rest of the factions, or invest enough into infrastructure to change into Free Market mode. With the Spartan +1 Police bonus Police State Free Market only drops you to -2 Police, so you don't have to worry about any extra drones from military units.