The OP sounds like a Hashkafah (outlook) question and not a truly Halachic one. IMHO There are various opinions in Halachah on how a Rav should deal with a person, based on their level of observance, in such a situation. An individual Rabbi might be strict or lenient, depending on the (perceived spiritual) level and situation of the person asking this question. So someone should consult their Rabbi for a ruling.
My answer will focus on my experience in Hashkafah with this type of question.
When I was in high school, R' Akiva Greenberg of Vishnitz taught me that one aspect of the benefit of Shabbos is that Hashem gave us the gift of being in control of our own life. We could be slaves, but Shabbos frees us. He gave an example: Most people will by habit answer a ringing phone. The phone rings and we run so we don't miss a call. Its "important". Shabbos comes in and says: "Nothing is more important than owning yourself." So for instance on Shabbos the phone rings and we realize it doesn't matter at all; because our lives are more important and whatever it is can just wait. There is a G-d watching over us and nothing is that important (unless its saving a human life; which G-d has asked us to be involved with Him in case it comes up. This is not because G-d "needs us" to save someone, but because He wants us to do what's truly important).
He told me that when he was a boy, Rockafeller center's building would close for business one day a year (which did not fall out on any holiday at all). The reason the owner gave for this was: "I want to show that we own this building, and therefore we decided one day a year we close."
The most important way to exercise "ownership" and to "be your own boss" is to exercise the power of NOT using something. Shabbos is the gift that teaches us just that. Are you proud of your hobby, or does your hobby own you? If your favorite show is on, do you have to watch it? Really? Why? You may want to watch it, and it may be good to have fun sometimes and enjoy life by tasting all kinds of good things and experiences... but not if you MUST, or else you will have withdrawal symptoms or resent missing it. Then it owns you. Really, you need to own it.
Shabbos teaches you that your time, your friends, your family, and your own growth and ability to own your mind and heart, are of the greatest importance.
Therefore, the teaching is that even if the game fell out on a Wednesday night, its still ok to "miss it" if you realize you do have something of greater value to accomplish instead. OR You may watch the game and enjoy, as long as it was your choice that time and you can easily miss the next one if you need to; because you are more important than any game.
The OP says: "those that will have their Shabbat experience ruined by not being able to watch the game"
Actually, no ... They will have their Shabbat experience ruined if they DO watch the game! Their game experience will also be more enjoyable if they willingly detach from it for more important things. That way when they do watch, they own the game time; so its more enjoyable. This is because enjoyment from choice feels more satisfying than enjoyment from fulfilling addiction.
BTW, the OP says: "and will be Plutzing for the last 6 hours of the day because they don't know the score?"
So since the case is that they can spend Shabbos afternoon without having non-Jewish neighbors run around informing everyone in the streets about the score, then not knowing is fine. Just pre-tape, the game, or watch a re-run on the internet after Shabbos, without asking about the score. Then it will be just as if you watched it a few hours earlier, just without all the commercials!
Get a designated person to volunteer to take care of recording it and have a Sat. night party at his house with pretzels and beer.