The obvious weaknesses in your friend's approach are relying on thin ground troops to hold subjugated systems and leaving the probe stack fat and the lack of opposition for mission cards.

I'd try focusing on ground troops/ground combat and associated missions. Constantly use missions to reduce their ground troops. Stack your own in the rebel base. Use cards to move them around. Keep at least one administrator safe for moving the rebel base, two is better. Use Infiltration to find objectives that work well with this strategy -- Crippling Blow, Decisive Victory, Liberation etc.. Toward the end, lean on target marker objectives toward the end of the game. Relentlessly sabotage, even if they always oppose, especially in ground unit generating systems -- loyal first, then subjugated. In subjugated systems this will keep their administrators getting plans, but not building anything and building loyalty...IOW spending two for one. Those are your first two priorities -- getting good objectives and focusing on the ground game. With what you have left, go for loyalty.

The idea is to starve their fleet of ground troops to give them a soft underbelly that you can cut into when the right cards come up -- Lead the Strike Team, Incite Rebellion, Assault, Plant Explosives etc.. When their fleets get close to your base, you'll have a good chance to fight it out for a couple of rounds then jump systems when they have you weakened using two admins and to pick from 8 systems.

Keep in mind: subjugated systems with rebel loyalty still count toward totals for the objective cards. They can't easily gain loyalty in subjugated sabotaged systems to remove sabotage and prevent you from gaining loyalty in neutral systems or subjugated unsabotaged systems.

Push those objectives. Make them regret spreading so thin. Make them regret giving you missions for free.