If your sensitive logs end up in the webserver root because one thread used chdir to temporarily change the working directory it's on the application writer.
Or to put it another way, the filesystem as a whole being shared mutable state does not make the current working directory being shared mutable state between threads any less of an issue.
As well as absolute paths, it’s ok to work with descriptor-relative paths using openat() and friends.
If one thread is using relative paths, and another is doing a chdir-based traversal (as using the nftw function, for instance), that first thread's accesses are messed up.
This is why POSIX now has various -at functions; the provide stable relative access.
The former allows you to design a coherent system. a lot of design questions which are annoying (“how do I access config data consistently, etc) become very clear.
It also makes C more productive. If global vars and static locals are unbanned, features like closures become less important.