EDIT: it also appears asymmetric. I doubt a French court can issue a warrant that forces OVH to turn over data stored at their Canadian data centre (similar with OVH and Finland).
On the other hand: GDPR.
All countries try to pull the same moves on the internet, from Russia to Pakistan to the US and Europe. The difference is that a small market has little leverage. The EU already has data-retention laws that could be extended, and at that point any EU government would have quite a lot of leverage. It's politically unpalatable for now, but things change.
It generally doesn't grant access to prosecutors, it tries to protect consumers and can lead to fines in case of violation.
Similarly, a US company could relocate to Iceland and thumb its nose at the PATRIOT Act and so on, but then "good luck doing any business" in the huge US market. The principle is the same: big markets can impose rules that more or less the entire world ends up following.
Its important to note that United States v Microsoft involved a US company (Microsoft) controlling and storing data (emails) on behalf of a US citizen. The emails were physically stored on a server that was located in Ireland. MS argued that a US judge doesn't have the authority to issue a warrant for data that is stored outside the United States and the FBI needed to go through cross-boarder channels.
I haven't read the bill but everything I read about the CLOUD Act seems to indicate that the act is meant to apply to US companies and US persons, not to foreigners as a rule. It's an extremely unjustified jump to conclude that "all data stored at any of the large cloud providers fall[s] under US jurisdictions." Unless you have more information than me that you aren't sharing.
The real leverage it has is the base of US users.