Honestly "master doesn't pass tests" is just a naming problem. Solving is just a matter of renaming master to something else ('dev'? 'potentially-incorrect'?) and making a new version of it that does pass tests.
And obviously, under the hope that somebody notices the failure by then, tracks it down to the right commit, and fixes the commit, instead of adding a workaround that avoid having to find the root of the problem.
"master doesn't pass tests" isn't just about naming. Its about culture. Its about agreeing that there is some source of code that everyone agrees on should work, and having the culture of agreeing that all modifications to it should work as well, and that it is the responsibility of those doing the modifications to make sure that's the case.
Continuous integration for compilers is a well-understood problem. There are posts in this thread explaining how Rust handles it. I understand that it's a royal pain to set up and maintain. I am a compiler engineer and am happy to work on compilation stuff but wouldn't want to touch our CI system with a ten-foot pole. But LLVM is a project driven by Google and Apple, there are really no excuses for not finding the people willing to do this.
If the commits don't even build they might as well be thrown away. It's going to make regression tracking next to impossible otherwise. The only reason to keep a branch around is because it's a small ephemeral branch under active development, or because it's an eternal branch that you can use to track regressions (using git bisect, for example).
But how would you tell which one that is? I guess even within the current system one could set up a bot that monitors master, and whenever the latest commit builds and passes tests, it updates a "known working" branch. But this doesn't seem to be the case. And if such a system is set up, it would make a lot more sense for contributors to push to a "testing" branch instead of master and then only updating master for working builds, rather than the other way round.
Sure, one way is to have an automatic test suite and an integration system that moves commits that passed it from one branch to another. If you can actually make a test suite that has no false negatives and finds enough negatives to be useful, that's great.
In my (limited) experience test suites are not close to that ideal. I've worked with systems where there was an automatic build test at least, and sometimes that catches a bad commit. But I think for many fast-moving projects there are few useful errors these systems can find, and sometimes they have annoying false negatives. The thing that I can think of that would most improve my productivity right now is a check that nobody added tab characters in the source code.
(I realize my comment might not apply to a situation like LLVM, where there could be a large test suite of programs to compile and run for example).
Compilers are also often very modular. LLVM in particular makes it very easy to take a bitcode file, apply a certain well-defined set of transformations, dump the resulting bitcode file, and check whether it fulfills the expected properties, e.g., "this computation has been optimized to this other computation". LLVM has a tool called lit that does this kind of test, and there is an extensive test suite that uses it. Of course none of this guarantees that other programs will behave differently, but in my experience these test suites really are very good and useful.
The only missing piece in LLVM is consistent enforcement of a policy that tests must pass in order for the repository to reach certain well-defined states.