I guess I'm somewhere in between on that spectrum. I use clang-tidy and valgrind constantly, along with an everchanging set of profiling tools that occasionally find a potential bug as well. I've used cppcheck in the past, but didn't find it particularly useful, and I use whatever VS2019's static analysis tool is called now if I'm on Windows. I'm also extremely pedantic about compiler warnings, in that I either have a specific reason for disabling a warning entirely for that codebase or it will get fixed. Exhaustive (often mathematically exhaustive) tests go on top of this.
That being said, I don't use any of the sanitizers in my usual workflow. The reason is simple - most of them don't offer anything I need. The huge set of bugs around excessive dynamic memory allocation simply doesn't arise when you don't do dynamic memory allocation, so ASan/LSan/MSan don't give me much. UBSan might theoretically help, but only with issues that have so far been very trivial to debug (accidentally misaligned pointer in explicit SIMD code and the like, results in instant crash and an immediate fix). The one exception is the ThreadSanitizer and when I happen to finally get a non-trivial deadlock, that'll certainly be my first stop. As it is, most of the multithreading I've done recently has been fairly trivial from a compsci point of view (and for good reason, overly fancy algorithms aren't often particularly performant on current architectures).
Regardless, I'm entirely open to the possibility that I've neglected a tool that might help me further. Strong tooling is one of the major advantages of C++ and I've tried most of the major tools at one point or another. I do require demonstrable benefit in the context I work in to continue using them, however, and I've often found it worthwhile to simplify my workflow by cutting something out that turned out to be mostly superfluous.
This is a personal quirk, but I consider C++ a different language from C++ + a lot of static analysis tools, on the grounds that most of my major beefs with C++ (and C) can largely be addressed by using powerful static analysis tools from day one. So from my personal, idiosyncratic perspective, I see two very different situations if the original poster is "using C++" or "using C++ and a suite of quality static analysis tools".
I prefer languages like Rust that take advantage of our decades of experience with C/C++ and write the most important stuff directly into the language, but in 2019 I still concede that "C(++) + good static analysis" is still a very economically compelling language for a lot of companies with existing C/C++ codebases. "Upgrading" to C+++SA is not cheap, but it's cheaper than any of the other options, as much fun as "rewrite in Rust" may be.
As for their belief that their code is thread safe, if they are using C++ and not C+++SA, I echo mike_hock's comment. It may be threadsafe, but from this distance and without a lot more evidence, my Bayesian priors say I don't believe it.