[ -The Solomani Government- seems not to do much with the opportunities presented, in canon.
I pulled my old Traveller books out of my oversized closet ... err, garage.
Alien Module 5 Solomani and
Rebellion Sourcebook confirmed my recollections. In both cases, the fleets at the 'tops of the horseshoe' attack Coreward, getting no closer to the announced goal(s), Terra and the old Solomani Sphere boundary. This reminded me of the Stalingrad / Caucasus campaign: develop a lengthy flank and leave it hanging. It also means the Solomani are trying to operate three separate theatres simultaneously. While knowing they have a limited resource base and will be overwhelmed if this war goes on too long.
I also noticed the Depot map, which I had forgotten about. Depot / Diaspora is right near the limits of previous Solomani claims. Hmmm...
The Solomani Grand Strategy should have been to cut off Vega (and Terra) and provide the Solomani government with real depth, THEN move Coreward. Initially, a pre-emptive demolition of Depot / Daibei heralds pincer drives Spinward and Rimward along the Diaspora / Solomani Rim sector boundary. Their secret final goal is Depot / Diaspora as a forward base for the next generation's war, but publicly they want to restore the Solomani Sphere (and maybe create some buffer states in between the Sphere and the Rifts).
More intelligent diplomacy would have encouraged Vega to secede from the Imperium, holding out the offer of the same autonomous status within the Sphere. If possible, wreck Depot / Solomani Rim because it's right next door to them and doing so advertises "The Imperium cannot protect you any more". For all the Imperial-Occupied Territories, downplay the Solomani Supremacy meme, instead proclaiming "we can drive away the fear and the chaos and the uncertainty. Join us!"
I found an old discussion thread of a Solomani client-state having an improved General Staff to help create 'genius generals'. I'll borrow that but mix in some IRL history: the IMTU genius who recognizes the Old Expanses opportunity and comes up with a Schlieffen Plan (now aiming directly at Depot / Diaspora) is actually a Staff Officer, not the visible Admiral in charge.