Rust was absolutely designed for programmer productivity. Source: I was one of the designers.
> GC is incredible productivity boost which is why no recent language (except Rust) does manual memory management.
That's true, it is a productivity boost. But it also has a cost. Rust didn't want to pay that cost, especially when the same mechanisms are useful for preventing data races. (In fact, removal of GC was something that happened as a consequence of the data race freedom features; it wasn't a design goal from the beginning.)
> Compile times in Go are a fraction of Rust compile times, which is another productivity boost.
And rustc performs much more optimization. Again, a tradeoff.
> Go gives you one good way to do concurrency, Rust believes in tyranny of choice.
That's untrue. Go has threads, channels, mutexes, condition variables, and atomics. Rust has threads, channels, mutexes, condition variables, and atomics. The only thing I can think of that Rust has that Go doesn't is, like, SIMD, and I'm sure your comment wasn't just referring to SIMD.