But this isn't an example of gender-neutral-singular-they. It's the perfectly usual problem where one group is several things, which are one group. Whether the group acts as one, or act separately and plurally, is entirely a matter of context and not of grammar at all.
Here the winner of the whole is a single golfer, who by synecdoche is representing his team, who by synecdoche are referred to with the name of the country they represent, which has a plural form but is a single country; his opponents represent a resolutely plural group of countries under a singular name.
If these things bother you you should probably avoid speaking any language at all. It's how the world is: don't blame the grammar.