Minnesota Fats was the real tragic character

Yes, but that's what builds "character" (as Burt says).

I think that when Eddie came back it was already known that Gordon staked him on the road, so Gordon really couldn't take a side, having also been staking Fats. This last match was just the two of them, their own money. The showdown after the second match between Gordon and Eddie where Gordon asks for half, then desperately negotiating for less than half.. but he settled for none of it so long as Eddie never plays "big time pool" again. This does imply that he has a piece of Fats, and he was willing to trade it for a piece of Eddie. If Eddie stayed on, Fats might have become free of Gordon. That may be why Fats suggested Eddie to pay up. And so I agree with some of the posts above, Fats is a tragic figure, trapped in the hustle and with Gordon.

You weren't supposed to realize how much Fats was in Burt's pocket until the end. We are supposed to see him from Eddie's perspective - the big cheese who he's there to slice up and replace. But there are subtle clues that Fats can't bet too deeply without Burt's approval. When he sends the preacher to get liquor, that's actually a signal to get Burt because Fats realized he was in trouble and shouldn't proceed without Burt's approval. He says something like, "get it from Tony's" meaning, swing by where Burt is playing poker and alert him to the situation. Then Fats won't continue to pay without a nod from Burt. I didn't understand what that meant until Eddie went through the same process.

At the end when Fats had been defeated, he sits there like a broken man. Or like a man who's been bought and paid for. He's so accustomed to living under Burt's rule, he can't figure out how to break away. And he looks even more pitiful when Eddie fights his way out.

Also, the thumb breaking was at Burt's instruction. Eddie finally figures that out in the end.

I thought it was pretty obvious that Bert owned Fats. Sure Fats dresses well, he hadn't lost in 15 years and probably taking all comers.

He doesn't necessarily "pick his own matches" either. Fats has to look to Bert for the OK for example, has to look to Bert to continue his match with Eddie the first time when he was down $18,000. Bert was obviously the one in control.

There were absolutely no false assumptions! The whole point of the movie would've been lost if that was the case!

Eddie threw Sarah over- TWICE. Once, when he publicly rejected her after she tried talking sense into him during his match with Findley and again when he told her to leave. Eddie, himself, admits it when he says "I traded her in for a pool game."