"Fails" is a higher-level concept than the shell is concerned with. Failure conditions and reactions are entirely at the discretion of the programmer and are not built as an assumption into the shell.
The only thing /bin/false does is return 1. Is that a failure? No, that's how it was designed to work and literally what it is for. I have written hundreds of shell scripts and lots of them contain commands which quite normally return non-zero in order to do their job of checking a string for a certain pattern or whatever.
Programs are free to return whatever exit codes they want in any circumstance they want, and common convention is to return 0 upon success and non-zero upon failure. But the only thing that the shell is concerned with is that 0 evaluates to "true" and non-zero evaluates to "false" in the language.
It would be pretty inconvenient if the shell exited any time any program returned non-zero, otherwise if statements and loops would be impossible.
If a script should care about the return code of a particular program it runs, then it should check explicitly and do something about it. As you linked to, there are options you can set to make the shell exit if any command within it returns non-zero, and lots of beginner to intermediate shell script writers will _dogmatically_ insist that they be used for every script. But I have found these to be somewhat hacky and full of weird hard-to-handle edge cases in non-trivial scripts. My opinion is that if you find yourself needing those options in every script you write, maybe you should be writing Makefiles instead.