If I understand correctly, then anyone with certain authority can file a DMCA takedown notice, which results in the removal of protected content. Anyone who posted the content and wants to challenge that notice can also do so, resulting in the restoration of the content, but not until a few days later.
In the long run, this mechanism does not suppress free speech or prevent the information from being distributed if proper counters are filed. However, in the short term it can delay that distribution. This is still potentially significant, but it is a different situation.
The position of this article and the lawyer it mentions seems to be that the "window" here is a problem, and should be removed. But isn't it fairly normal in matters of legal dispute for courts to make temporary orders that leave options open as much as possible while the case is heard and a considered decision can be made?
In the Internet age, any interesting information that gets out tends to stay out and spread within a matter of hours, so it seems to me that if you allow an immediate counterclaim to be filed and the information to be restored without delay, this will in practice negate the benefit of the whole DMCA takedown process and make copyright protection fairly pointless as realistically it will be unenforceable.
Of course, there is also the balancing point that other kinds of information that have only short term relevance could be censored for long enough that they also become worthless, as in the election case.
Fundamentally, I don't think you can ever solve this problem completely unless you can have a proper court decision made within hours of detecting allegedly infringing content. It's not reasonable to expect content hosting services or ISPs to become judge and jury.
That being the case, I wonder if the current situation, where material can be temporarily blocked but only for a few days, really is the best compromise. That allows time for all parties concerned to make representations to a court, and the court should certainly be able to penalise anyone who it determines was deliberately abusing the system either way around. But at least the right (as in, court-ordered) long term decision about the release or containment of the information will be possible in most cases, and there is a limit on the amount of damage that can be done in the short term. Given the potential damage in cases like these once Pandora's box is opened, and the relatively limited damage that is likely if release is delayed for just a week or two, is that such an unreasonable compromise?