I haven't done the research but I thought I heard the reason he wasn't eligible for FedEx is because he isn't officially on the PGA Tour. He's been getting sponsor exemptions and qualifying based on his performance but he still isn't a member of the Tour. He's playing so much because he's getting the exemptions and he's trying to win one. My understanding is the only way for him to qualify for this year would be to win an event otherwise he was to wait until next year to play FedEx.
NFL, NBA and NHL are totally different with their own rules for eligibility within their leagues. He couldn't play in the NFL because he's 19. There are probably only a few players on MLB rosters that are 19 and I think the NBA uses years of college ball and / or age as their criteria. Plus each sport has a cut off date that a player has to be on the roster for in order to play post season.
I don't know how tennis works but that would be the closest to the PGA Tour in terms of how players qualify for Tour events.
I think you are correct as far as I can tell, BUT he is a "Special Temporary Member" and can get into tournaments not only through sponsors exemptions, but also from category 31 of the Tour Priority rankings, prior week top 10 and Monday qualifying. I believe the minimum age for a tour member is 18- even if he was 35, he would not be playoff eligible without a win which would then make him a full tour member (which he is guaranteed of being next year based on this years results. Bill Haas was able to win the Fed Ex Cup with no wins and 6 top 10s during the "regular" season. Spieth already has 5 top tens so I don't see way winning alone should change things for him.
I think team sports have cut off dates to avoid late season trades that stack playoff bound rosters. In any case, Spieth has been a "Special Temporary Member" since March, way before any equivalent cut off date (and would have earned enough points based on his results since then even if you exclude his pre ST Member results.