(Original poster here)
A quick followup. I got together with one of the people in our original foursome, and we played for about 8 hours on Sunday.
While looking up some rules to a different question, my friend noticed one major thing I had gotten wrong, which basically invalidated all the games we had played to date: we had been giving cards in the Squad Leader position +1 armor vs. every attack, not just the first attack of the turn. Obviously this makes a huge difference.
We’ll be getting together again soon, and I’m also taking Codex to San Francisco when I visit some gaming friends for a long weekend later in the month, so I’ll have a new group of people to play with.
Oh, also, the suggestion of turning the discard pile sideways and putting the tech cards on top perpendicularly helped me a lot.