For example: embedded applications, or when I'm the only one running trusted code on my computer. Think also of scientific computing.

If you insist on networked solutions: database servers.

Also: "multicore + purely functional style" is just as clean as "multiprocess + purely functional style", but ... the latter is less efficient because of communication overhead.

Unless you manage to do any kind of work without IO, there is always a security risk.

And if that is irrelevant to you, there is still the application stability and memory consistency to ensure data is being handled in a memory consistent state.

Fortran is a good example on how to do scientific computing without low level threads.

Yes threads should be used with extreme caution, that's why I advocate using a functional programming style. But you still want the performance of multicore. It's not one or the other (but with OCaml it unfortunately is).