This is an organizational problem, of course. Rust can be different things to different people, without necessarily forcing one group to compromise overmuch. But some tension is probably inevitable.
That depends on the language. In an interpreted language (including JIT), or a language that depends on a dynamically linked runtime (ex c and c++), it isn't directly included in your app because it is part of the runtime. But you need the runtime installed, and if your app is the only thing that uses that runtime, then the runtime size is effectively adds to your installation size.
In languages that statically link the standard library, like go and rust, it absolutely does impact binary size, although the compiler might use some methods to try to avoid including parts of the standard library that aren't used.
What you say is true enough for external-runtime languages and Go, though TinyGo is available for resource-constrained environments.
The no_std Rust only has core but this is indeed a library of code, and freestanding C does not provide such a thing = freestanding C stdlib provides no functions, just type definitions and other stuff which evaporates when compiled.
Two concrete examples to be going along with: Suppose we have a mutable foo, it's maybe foo: [i32; 40]; (forty 32-bit signed integers) or in C maybe they're int foo[40];.
In freestanding C that's fine, but we're not provided with any library code to do anything with foo, we can use the core language features to write it outselves, but nothing is provided.
Rust will happily foo.sort_unstable(); this is a fast custom in-place sort, roughly a modern form of introspective sort written for Rust by its creators and because it's in core, that code just goes into your resulting embedded firmware or whatever.
Now, suppose we want to perform a filter-map operation over that array. In C once again you're left to figure out how to write that in C, in Rust foo impl IntoIterator so you can use all the nice iterator features, the algorithms just get baked into your firmware during compilation.
They also have a lot narrower scope of use, which means it is easier to create stdlib usable for most people. You can't do it with more generic language.