Unfortunately micro-architectural side-channels are much, much more than leaking information through timing of victim code, and I thought by 2021 this would now be widely known. For one thing, Spectre variant #2 can allow attacker processes to inject speculative information leaks by polluting the BTB to induce essentially any speculative behavior you want. That renders all compiler-based mitigations basically ineffective, because an attacker can just bypass them.
There are other microarchitectural leaks like L1TF that allows (on some unpatched, unmitigated CPUs/kernels) the ability to speculatively read all of L1 cache, regardless of privilege level. Let alone speculative attacks on the kernel aimed at reading physical memory, which bypasses any in-process you could think of. And of course, this paper doesn't address the fact that memory access patterns are not affected, so the classes of attacks that reverse address translation and ASLR are on the table. I didn't read the paper in detail, but there's always the problem of mis-classifying "secret" data and not applying mitigations everywhere.
Unfortunately, the authors misuse of terminology will cause this paper to be misunderstood. It has much, much narrower applicability than it would appear. It only applies to specific kinds of timing channels. This might be worthwhile to apply to codes like crypto, but honestly, those need to be designed much more comprehensively now, rather than just applying a compiler analysis to the problem.