Auto merge of #142002 - onur-ozkan:follow-ups2, r=jieyouxu

redesign stage 0 std follow-ups part2

Fixes three bugs:

1. `x check` fails when run on rustdoc without `download-rustc` enabled. (1st commit)
2. `x check` fails when run on the compiler with `download-rustc` enabled. (2nd commit)
3. `x test library` fails with `download-rustc` enabled. (3rd commit)

Fixes rust-lang/rust#142018 (case 1)
Fixes https://github.com/rust-lang/rust/issues/141983 (case 3)