The attackers don't have control over the observed address, so they can't easily read arbitrary memory, but careful scanning for the type of data you're targeting can still make the attack effective even against the OS kernel. For example, since URLs can be picked out of memory, this apparent proof of concept shows a separate process running on the same CPU victimizing Firefox to extract the URL as the user types it in. This works because as the user types, the values of the individual keystrokes go through the LFB to the L1 cache, allowing the malicious process to observe the changes and extract characters. There is much less data available to the attacking process but that also means there is less to scan, making real-time attacks like this more feasible.
That said, because the attack is specific to architectural details of HT (and the authors of the attack say they even tried on other SMT CPUs without success), this particular exploit wouldn't work even against modern high-SMT count Power CPUs like POWER9. It certainly won't work against a Power Mac because no Power Mac CPU ever implemented SMT, not even the G5. While Mozilla is deploying a macOS-specific fix, we don't need it in TenFourFox, nor do we need other mitigations. It's especially bad news for Intel because nearly every Intel chip since 2011 is apparently vulnerable and the performance impact of fixing ZombieLoad varies anywhere from Intel's Pollyanna estimate of 3-9% to up to 40% if HT must be disabled completely.
Is this a major concern for users? Not as such: although the attacks appear to be practical and feasible, they require you to run dodgy software and that's a bad idea on any platform because dodgy software has any number of better ways of pwning your computer. So don't run dodgy programs!
Meanwhile, TenFourFox FPR14 final should be available for testing this weekend.