I've been playing nearly a game a day (on normal) for the last 2 weeks, and had a miracle run on normal last night.
I was gifted the Glaive beam after a jump or two, and quickly upgraded my weapons to take advantage of it. I hadn't been using beam weapons at all, but the description was such that I decided to forgo my usual upgrade path and sink a bunch of cash into the weapons. It worked very well. The Glaive beam tore through most enemies until around sector 5. I had held onto the BL2 and Artemis just in case so having those on standby was useful for the more shielded enemies.
Being able to tear through enemies means I wasn't taking as much damage as normal, and had enough scrap to upgrade as I pleased, and tour around looking at as many beacons as possible.
After a brief stop at a repair depot in the final sector, I made a bee-line for the Flagship. With a full arsenal, cloaking, teleport and a full crew, I was as confident as I'd ever been.
The first stage was tedious but not difficult. I had a small number of missiles, so landing shots was difficult.
The second stage required more management but even the attack drone wasn't causing too much trouble.
The third stage was a bit of a surprise as I wasn't expecting the super shield and was worried by the missile volley. Regardless, it was dispatched soon enough.
After about 30 attempts it was pretty nice to finally finish the game, even though I had some ridiculous fortune on the way.
For now I'll probably bask in the victory for a few days, then try out some of the ships I've unlocked along the way.