After I have defined a high-performance algorithm, they often want to modify it to make it "more beautiful" (which means functional) and in the process destroy the performance.
That's annoying. They should understand the concept of interface as separate from implementation. Implementations are allowed to be (and sometimes should be) ugly if performance concerns merit it.
Besides, mutable state, intelligently used, isn't "ugly". A lot of programs are simpler and more attractive when mutable state is used. Mutable state just needs to be used with taste.
What really pisses me off about that behavior pattern is when the tinkering is unnecessary. If it's working code, then who cares? I'm a hard-core FP advocate but I write stateful programs all the time. I try to wrap them behind interfaces that are as functional as possible. I will use state in controlled ways when appropriate; I just don't want to impose it on other people.
I work on low-level algorithm design in A.I., where performance is everything and there are many interlocking, low-level and complex algorithms and data structures.
I think what makes C++ appropriate for your purposes is that once code is written to spec, it doesn't need to be maintained (except for performance optimizations). I'm guessing that (except possibly for some STL usage) you're actually C.
Where C++ and Java really drop the ball (legacy disasters) is when code has to be read more times than it is written. If your goal is to write fast software one time and, once it's working code, never need to look at it again, C++ is probably appropriate.