I think it's usually fine to use the word "points" instead of "goals", as they often mean the same thing. I'd still not use it with this sentence, because the way of scoring the points (goals) is specified. There was anyway a small mistake in your answer (or was it just a typo :'D); it was in past tense (the original sentence was in present). So according to your comment your answer was (when translated to Swedish) "Spelaren GJORDE fyra POÄNG". If the "scored" wasn't a typo, the correct answer is here: "Spelaren GÖR fyra MÅL".
Sorry for the veeeeery long answer, but I hope this helped you :)
Maybe you are right in some cases butit is not exactly the same meaning EX. when you score a goal in basketball it can be 1or2or even 3 points and in american football it can be more points.i think if we could use other words instead of score it could be helpfull to get the better considration