Auto merge of #29895 - mrobinson:cleanup-rustdoc, r=jdm

Clean up rustdoc run

1. The options specified in `rustdoc-with-private` are essentially the default now so we can remove this script.
2. `./mach browse-doc` is redundant with `./mach doc --open` which uses the underlying cargo functionality to browse the documentation.
3. `etc/rustdoc-style.html` is unused and can simply be removed.

Fixes #29888.

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- [x] `./mach build -d` does not report any errors
- [x] `./mach test-tidy` does not report any errors
- [x] These changes fix #29888 (GitHub issue number if applicable)
- [x] These changes do not require tests because they just update build scripts.

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