Can the service providers somehow block illegal streaming themselves? That way no third party services would be affected?
The situation is a bit irregular, as the streaming providers set up a new website for each game, and the legal system isn't fast-moving enough to issue a court order banning a website within the 90 minutes of a football game. Instead La Liga got a 'dynamic blocking injunction' so they tell ISPs what to block, and ISPs have to block it.
Then they can just ban this customer. That way the authorities will not have a reason to ban IP ranges affecting the other customers.
But aside from it, it should be very obvious: A) you are notified by the intellectual property holders that somebody is streaming pirated content, B) a specific customer or set of customers, who are not a known streaming service, are serving tens or hundrends of IPs with video and C) these customers do not have much activity during other times.
Plenty of people stream commentary to matches without showing the game itself, so that would flag as guilty too
In theory someone might rent a server and do the streaming directly to his viewers, without using a known platform. This would be a legitimate false positive as you describe. But this would be so expensive I doubt anyone would do it when the alternative is a free platform with built in community and monetisation tools.
Cloudflare is not ignoring LaLiga and they are open to collaborate, but LaLiga refuses to do so, and are battling legally over it.