This will indeed mean that when a person claiming copyright ownership has filed a DMCA takedown notice, and the alleged infringer has filed a counter-notice, the claimed copyright owner will be unable to file a suit within 14 days unless the copyright was already registered, except in those cases where the author is not a US national, in which case the requirement for registration does not apply.
Of course, the US Congress could amend DMCA to change the takedown procedure after a counter-notice.
However, it should be noted that counter-notices are much less frequent than are take-down notices, and that suits are filed within 14 days in only a small fraction of counter-notice cases. It should also be noted that an ISP or other hosting provider is not required to restore the content after a counter-notice if suit is not filed within 14 days. The only consequence for not restoring the content in such a case is that the provider loses its shield against liability for removing content. If the poster can show damage, and can also show an obligation to host the content, s/he could sue the provider for failing to restore the content. But in the common case where the provider has no duty to host any particular content, or has reserved the right to take down any content at any time for any reason, there would be no liability, and so no penalty for failing to restore content. In practice, only where a provider is required by a contract to host the content is a suit plausible, and then only if the poster can show damage from the content being taken down. So in practice, there may well be little change.
The Wikipedia article on the case says:
Others have observed that the practical effect is minimal, since although the turnaround time for a registration with the Copyright Office can be months at a time, the Office offers an $800 expedited review process for cases with "compelling needs" like upcoming lawsuits, and offers preregistration for categories of works more likely to be infringed. Moreover, plaintiffs can recover for any losses accrued from infringement, even while waiting for the examination to be complete.
The decision applies only to filing copyright litigation, and does not apply to other forms of enforcement, such as sending demand letters or issuing DMCA Section 512 takedown notices.
In an article about the case posted on scotusblog.com, Jessica Litman says:
The opinion resolves a longstanding circuit split, but the practical implications may be modest. The decision may encourage some copyright owners to register their claims promptly, may reduce forum shopping, may delay some infringement suits for several months and may deter plaintiffs from including peripheral copyright infringement claims in suits over other disputes in order to brandish the threat of large copyright damage awards.