First, C++ programmers in aggregate have to consider more than the choice of language for their projects. Odds are they work for somebody else, on a team, an established project, whatever. In these cases "C++ programmers need better tools" doesn't mean "toss the bath out with the water." Rust isn't necessarily a good move in established products, in other words.
Second, Rust would do better, in my opinion, to target folks who are new to systems programming and, for whatever (mostly invalid, in my opinion) reasons, are intimidated by C or C++. Rust is very powerful and in many ways I like it much better than C++. I think Rust offers a great way for these people to have a gentle introduction to the world of systems programming. I say "gentle" because a good portion of the "pain" of learning is front-loaded into getting a build, rather than back-loaded into futzing about with gdb (although that would be something these folks will miss out on--who doesn't like a good session with gdb with a satisfying bug squash at the end?). Get them while they're young, as they say.
The overhead for existing C++ teams is learning the language, and (potentially AFAIK) interfacing with C++ code that has no good C style API. But Rust seems to tackle even the latter quite well. I don't think you'll get around the learning curve for a new language.
This is interesting. Many universities teach Java, C, C++ or Python as a first language. The headache that many first year students suffer from these basic (relatively speaking) languages is readily apparent. Rust is very strict, and it requires an entirely different mindset to use. I wonder how this would fare with students who have a clean slate.
I suspect that the % of students who have difficulties won't be much different, overall. There may be a different distribution of when they run into trouble.
The interesting question is whether the group of students who have problems with other first programming languages is different from the group that have problems with Rust-as-a-first-language...
In my experience, most developers can't or won't put in this sort of investment. Which leads me to wonder what fields/niches will Rust land in?
This might change if, as other commenters have stated, it is taught as a first or second language, but that doesn't seem likely anytime soon...