Side channel attacks are cool ways to leak information from a cryptographic system. They're often more about targeting the engineering and implementation of an algorithm as opposed to its design.
It's an area with subtleties where toolchains can introduce flaws in otherwise secure code. For example, constant-time string comparison is a common cryptographic need. And it's something that developers have to implement and review carefully because compilers love to optimize code -- and optimized code might introduce time-based side channels.
Also check out this article, which includes comments from NIST that indicate the attack doesn't appear to be a fatal flaw in the algorithm's design.
In the meantime, see how long it takes to upgrade your org's web footprint to TLS 1.3 and HTTP/2 (or, even better, HTTP/3). It's important to be prepared for new attacks, but there's a far more pressing reality of adopting new protocols in the first place.