When you find you're seeing a problem as easy, and that "they should just", then you probably either don't have enough detail, or haven't thought about an issue enough.

For example, what else should 'morals' compel Cloudflare to spend money on blocking? Should they preemptively drop accounts that link to any abortion information? Did anything happen at Tiananmen Square?

Let's set all that aside though and say "unauthorized streaming bad", and say that we all agree that that is the case. Say we also agree that it's Cloudflare that should be the enforcer of this. We then run in to the practical issues. Random numbers I found online say that Cloudflare streams around 100 petabytes of video every month. How do you propose that it filter that amount of video and identify the 'offending' streams? There's legitimate license holders that could be streaming through Cloudflare, cut them off and you're looking at a lot of lost business if not a lawsuit against you. There's clips being shown during a recap on someone's sports info stream that probably fit some 'moral fair use'. Both those rely on being able to distinguish between this current streamed game and a replay of a game for a year ago, even in cases where the streaming party has taken measures to make that difficult.

How much extra would you be willing to pay Cloudflare for their services so they could do this type of 'football stream blocking'? Are you okay paying the extra to Cloudflare when the end result is that there's no fewer streams available online?