CPUs have been vulnerable to this attack since 1995. How did it collectively take us 22 years to figure this out? I know it's a highly esoteric complex attack, but there's no shortage of clever hackers in the world.
You can have both "modern web" and block most JavaScript. You just need to keep adding scripts to the whitelist until sites you trust work again. It's a bit arduous at first, but possible to get used to once everything you visit daily has been added.
Unless, of course, the site you trust is hosted in a shared hosting VM which is also vulnerable to spectre or meltdown. In which case, you can’t trust the scripts.
The web issue is easier to mitigate if not fix completely since there is already a massive infrastructure for widespread, rapid browser updates, and crippling Javascript to eliminate attack vectors such as high-resolution timers is completely acceptable.
The cloud/vm infrastructure is a massive problem though. It is 100% required that VMs be fully isolated. The entire infrastructure breaks down if they aren't.
It's sort of been well-known that speculative execution opens up the possibility of side-channel attacks for quite some time. Hell, it's long-known that SMT (e.g., HyperThreading) can leak keys in a not-really-fixable way.
What's new and surprising is the power of these side-channel attacks--you can use these, reliably, to exfiltrate arbitrary memory, including across privilege modes in some cases (apparently, some ARM cores are affected by the latter vulnerability, in addition to Intel).
Honestly we knew about this in the 70s. Mainframe/time share systems had lots of protections against attacks like this. The problem is mainstream computing when cheap/single user and attempted to build a multi user/untrusted code execution environment on top of it. Now it's come back to bite us in the ass.