I suspect you were on the right track with reflowing the headers on the displays, but there's two sides to every connector! The likely culprit are broken spring-clips on the female-side of the connector. They can be difficult to spot, but a cracked or bent spring-clip can result in a poor/intermittent connection, and the kind of flickering you are seeing. To find this, isolate it to which connector it is first by unplugging all but one display. No flickering? Not that connector. Add in displays until you see flickering. Then, you'll have to pull and inspect each spring-clip (there's a tool for this, but careful use of a jeweller's screw-driver can release the spring-lock and allow removal).

I'm not saying that this is positively the problem, but it is worth a look: This problem affected my F2K that had 6 or 7 broken spring-clips, and upon replacement, the game was solid.