I use valgrind regularly, and prefer it over asan. asan will result in a faster executable which is nice, but I far prefer valgrind's output than asan's (this might be preference, but I find it to be clearer), and various things break when building with asan so I never make it default. Being able to valgrind stuff without recompiling is very convenient.
I'm also not sure if asan has an equivalent to --leak-check=full
Ok. But I'm guessing it has the same problems. I.e., if half your libraries/modules have never seen it, then you'll get a lot of noise. Happy to be proved wrong.
The compiler can't add checks into code it hasn't compiled. External modules, unless they are doing weird things which you do want to know about, should not generate ASan reports... on Linux.
I'm also not sure if asan has an equivalent to --leak-check=full