The real thing here is the definition of the Player type.
It should have a mutating method that resets it's score (a method defined on *Player).
Raw access on fields is often only ok for naive, trivial implementations. Methods are often needed anyway.
That's straightforward. Immutability doesn't make any sense here.
... Examples on how to avoid the bug ...
Notable is that you claim here is an opinion not an argument.
That immutability is not needed is also not an argument against it.
The point here is that working with immutables relieves you from some cognitive stress and repetitive work such as writing accessors and making fields private.
With mutability, it's still possible to make things immutable if properly encapsulated.
If everything is immutable by default and you need mutability for some reason, good luck.
The heuristic is simple too. If you need something to be mutable, you should deal with pointers. So *Player and []*Player here.
At the type/variable level, it's the opposite. We see it with const. Once it's in here, it's in there.