What does "Tree someone's or their game" means?

I'd say MW's second entry on the verb "tree"—

to put into a position of extreme disadvantage : CORNER
especially : to bring to bay

is the closest meaning that would make [any] sense in the context. The frame of the same verb here is being taken negatively, as opposed to your perceived context.

So essentially, "Instead of having treed their game..." could imply that instead of having cornered their game, or instead of owning their game, the gentlemen will still find themselves barking up the wrong tree (had they not done [something] in a specific way).

Both expressions likely come from English fox hunting. The hunter (usually on a horse) would have several dogs which would seek out and pursue a fox, with the purpose of tiring it or cornering it so that the hunter could shoot it. (Note that I may be a little off on the details of fox hunting here.)

If the fox chose to scamper up a tree it was said to be "treed". Usually the dogs would then circle the bottom of the tree, barking at it. But sometimes the dogs would mistake which tree the fox went up, or the fox would leap from the first tree to another, leaving the dogs "barking up the wrong tree".

While these terms are used metaphorically in many different circumstances, the origins have nothing to do with "games" that are "played" (eg, cricket). Rather, "game" refers to animals that are being hunted.