Conceptual consistency is a good thing, but there is a generally higher cognitive load to using C++ over C. I've used both C++ and C professionally, and I've gone deeper with type safety and metaprogramming than most folk. I've mostly used C for the last few years, and I don't feel like I'm missing anything. It's still possible to write hard-to-misuse code by coming up with abstractions that play to the language's strengths.
Operator overloading in particular is something I've refined my opinion on over the years. My current thought is that it's best not to use operators in user/application defined APIs, and should be reserved for implementing language defined "standard" APIs like the STL. Instead, it's better to use functions with names that unambiguously describe their purpose.
I still think being able to define a type that models what you're doing is incredibly valuable because as long as you don't step outside your type system you get so much for free.