I recently started a new game and I basically bum-rushed the nearest player and I'm just about to finish him off and it feels like a way stronger start than all my previous tries.
Boom. That is the key sentence. You are right. Pushing out fast is absolutely, positively, the winning path. Teching up simply doesn't work, or at least it's not ideal.
This is particularly true on that map, where if you ever move out, they take your cities, 100%. You just can't defend everywhere, because with 100000 teleporters, anyone on the map can get to any city you have in 3 turns, and they always know what your weak points are. Your only choice is to take out opponents.
So knowing that, I just did the campaign again, and was blowing it out by turn 43:
Two down, 2 almost down. I suspect a Turn 50 win +/-, based on where I was.
So what is the big difference? How did it get so one-sided? Looking at your post, the problem seems to center around getting the psi-fish out. I got out 6 by turn 12, which is when I got Friendship, plus the Negotiator award for reaching it first (see the events on the right side):
They looked like this:
That unit is CHEAP, and is decently strong. Virtually no investment. And the only reason it took this long is because I wanted the perk Energy Efficient Assembly Line and shield (landmark). I know how much that means.
Modding them out was a bit tougher than expected, though. A lot of the players on this map had already seen each other, so I couldn't rely on diplomacy sales from contacts. By this point, I only had 3 or 4 sales that way, which wasn't enough, not for what I was doing. So what then? Ditch a sector. Here is my first sector trade, on Turn 14, when my city hit 8 pop:
Annexed a worthless water sector, then sold it for 110 cosmite and 600 energy. Huge. I sold these 2 by Turn 20, plus all contact information:
Which bagged another 200 cosmite, and 100 energy:
It was just enough to mod everything out. I may have to sell 1 more soon.
So that is the money and plan, so how do you get the influence? Building up the influence relies on doctrines. Two of them require you to find the NPCs. There are three on this map: Spacers, Psi-fish, and Autonom. Find them and start filling those quests. I raced up to 5 total scouts, first thing, and sent them everywhere. Note that with your starting tech, you can see who-is-who in sensor range because of your starting technologies. This part is cake. Open them up.
Then get the doctrines up. There are three at this stage: 1. The Emissary 2. Noble Diplomats 3. Negotiator. 1 and 3 are free - you just have to finish quests. 2 is your first (T1) doctrine technology, which is the cheapest thing of all.
I got 1 and 2 up, fast as I could, then swapped between 1&3 when I was about to purchase the units. However you do it, keep as many of those up as possible. Bottom line is that you get 30 influence on each quest, with Emissary, and it costs 15 influence to buy a Chrysalis, if you have Negotiator. That is 2 chrysalis per quest, ideally. If you are doing quests for everyone, which is 6 chrysalis every 5 turns, PLUS my 17 (1.2 more) influence /turn with Noble Diplomats. With that math, the screen should be full of those guys by turn 30.
Again, buy Chrysalis only, then Hunters only, which will quickly do this:
This image was on turn 22. I had just gotten my third lord, who was heading fast (hoverbike), to join my clearing party. Nothing could resist my stacks at this point, and I was clearing a concentric circles towards an enemy.
I planned a timing attack sometime in the late 20's. I just wanted those Siren first. I cleared, and cleared, and cleared, then just as the Siren evolved, I took out Zetrochious. That was on turn 29:
No problem.
Note: I got those spacer hellcopters because their Mantra of the Branded is BUSTED. It was a jump ball between them and more Hunters, but I felt it was in the bag already. Plus, NPCs at integration no longer make demands. Whatever though, up to you.
The Siren looked like this:
Look at that upkeep. Isn't that beautiful? 6/turn for a siren. Gorgeous.
And I was right on him:
That is his capital and his lord. He is out on 29, and I have his cities too.
This is definitely the way to do it. So I did it again, over and over.
You seem to be getting it, though. The name of this game is "keep attacking." The xp on your units, and the rewards for fighting, make the cost of passivity WAYYYY too high. You seem to be acknowledging that though, so my guess is you will start every game now on your toes, leaning forward. That was how it happened for me, at least. Once I did, things really changed.