Not to be a nitpicker, but:
Remotely command semi-autonomous self-replicating mining machines to take over an entire asteroid belt.
Dyson Trees are trees designed to grow in an anaerobic environment (ie, no oxygen), with the express purpose of producing said oxygen, which would theoretically collect in pockets, making asteroids and moons and planets livable.
What you've got in your summary there is a Von Neumann machine. In fact, you might even have picked it up from Arthur C. Clarke's Odyssey Two (2010), where Jupiter was supercondensed thanks to a fleet of semi-autonomous self-replicating mining machines that sucked up hydrogen, converted it to neutron matter, and dropped it to the gas giant's core, causing it to implode and ignite under the weight of it's newfound gravity.
While Freeman Dyson did theorise a self-replicating machine, it was a machine, and not a tree. The two are mutually exclusive.
Which brings me to my second point (sort of). In strategy games such as this, you generally have 3 classes of unit. Offensive, defensive, and resourcing. The former 2 have been largely nailed, but not the third.
Given that the original design of the Dyson Tree was to flourish on barren asteroids and produce oxygen, maybe you could find a way of working this into the game. If seedlings colonise asteroids, produce 3 types of tree (spawner, defense and miner), you could either use that oxygen to create a new type of attacker, or even better, feed it back to the asteroid to improve the Energy, Speed or Strength stats. Ie, pick one, it slowly increments over time.
And if you could move that captured oxygen between asteroids, you could, in time, convert a section of a belt to produce exactly the kind of seedlings you need.
Just a thought