I think we are focusing too much on the game aspect of the system. There are likely a multitude of ways to "fix" FGITW, depending on what you view as the worst aspect of it.
But it only affects the first five or ten minutes of a given question.
As long as the answer is good within the short editing window, then the purpose of the site has been served - a question got an answer.
The purpose of the site isn't to gain reputation, nor is it to make certain that people get their reputation in the order they stood in line, read the question, or answered it. Those are important aspects of the system - they provide the motivation for answering questions - but they aren't the purpose of the system.
Right now, FGITW doesn't significantly reduce the motivation for people to participate, nor does it break the primary purpose of the site.
Further, people do get dinged when they post incomplete answers - they get downvoted, and sometimes flagged. The system does provide the means for others to make sure people know that incomplete answers are not acceptable. I don't participate in real time now, but when I did I found more often that what some thought of as "incomplete" were really good enough to solve the problem. I don't recall the last time I saw an answer that was truly just a placeholder, or didn't give some insight into the question.
If you see someone performing that sort of abuse, especially on a regular basis, flag it and explain to the moderator that the first edit truly was a placeholder, nt an answer, or misleading.
I suspect that the problem is small enough that it can be taken care of without changing the system.