Rollup merge of #135501 - tgross35:stdlib-dependencies-private, r=bjorn3

Inject `compiler_builtins` during postprocessing and ensure it is made private

Follow up of https://github.com/rust-lang/rust/pull/135278

Do the following:

* Inject `compiler_builtins` during postprocessing, rather than injecting `extern crate compiler_builtins as _` into the AST
* Do not make dependencies of `std` private by default (this was added in #135278)
* Make sure sysroot crates correctly mark their dependencies private/public
* Ensure that marking a dependency private makes its dependents private by default as well, unless otherwise specified
* Do the `compiler_builtins` update that has been blocked on this

There is more detail in the commit messages. This includes the changes I was working on in https://github.com/rust-lang/rust/pull/136226.

try-job: test-various
try-job: x86_64-msvc-1
try-job: x86_64-msvc-2
try-job: i686-mingw-1
try-job: i686-mingw-2

compiler/rustc_codegen_cranelift/patches/0029-stdlib-Disable-f16-and-f128-in-compiler-builtins.patch

@@ -15,7 +15,7 @@ index 7165c3e48af..968552ad435 100644
  edition = "2021"
  
  [dependencies]
- core = { path = "../core" }
+ core = { path = "../core", public = true }
 -compiler_builtins = { version = "=0.1.146", features = ['rustc-dep-of-std'] }
 +compiler_builtins = { version = "=0.1.146", features = ['rustc-dep-of-std', 'no-f16-f128'] }