Rust doesn't have the "static in header" problem, but it does have the "always inline" and "macros" problem.
The "always inline" problem is smaller in Rust, because it does Thin LTO by default, compiles partially to bit-code, that can be partially inlined if necessary, etc. So essentially the Rust toolchain is a bit better at inlining when its profitable than C and C++ toolchains "by default" (you can tune C and C++ toolchains like clang to be as good as Rust).
In C and C++, macros are generally frowned upon, not because of this compile-time issues, but rather because they are dangerous, tricky, non-hygienic, powerful usage requires complex patterns, their interaction with header files and PCHs, etc.
Rust macros are awesome, super useful, widely used, etc. So people end up using them a lot, and this is by design, not a flaw of the language. This ends up resulting in a lot of duplicated code being expanded, and that leads to the problem that you mention, but much worse, because there are just many more macros in Rust.
This kind of applies to templates / traits as well. There are many C++ that don't write templates, but all Rust programmers use generics because they are great. Rust can pre-compile generics, so the cost of this is not as bad as for C++ where the same generic might be instantiated by multiple TUs. But still, just by the fact that they are more widely used, the size of the problem grows.