why shouldn't the publisher also be responsible if they have proven they had the power to filter it out with mysterious algorithms and didn't?

The details seem a nightmare to regulate. What happens if they have a CP filter, but an item of CP makes it through? Are they then hit with the penalty for CP distribution? "As soon as possible" sounds impossible to litigate. I like CP as an example because Twitter doesn't want to distribute that (ostensibly). This isn't a matter of viewpoint filtering. This is a matter of law enforcement.

Having thought about it some more, perhaps there is more than one element of regulation. One goal is: you want pirated and otherwise illegal material taken down. The other is, I think: you want some fairness in moderation, lack of viewpoint discrimination. You can address these separately.