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

library/sysroot/Cargo.toml

@@ -1,3 +1,5 @@
+cargo-features = ["public-dependency"]
+
 [package]
 name = "sysroot"
 version = "0.0.0"
@@ -5,10 +7,10 @@ edition = "2021"
 
 # this is a dummy crate to ensure that all required crates appear in the sysroot
 [dependencies]
-proc_macro = { path = "../proc_macro" }
+proc_macro = { path = "../proc_macro", public = true }
 profiler_builtins = { path = "../profiler_builtins", optional = true }
-std = { path = "../std" }
-test = { path = "../test" }
+std = { path = "../std", public = true }
+test = { path = "../test", public = true }
 
 # Forward features to the `std` crate as necessary
 [features]