Rustup gets confused when it can't create a temp directory

Problem

$ rustup component remove rust-docs
info: removing component 'rust-docs'
info: rolling back changes
error: could not create temp directory: /acct/jynelson/.local/lib/rustup/tmp/ibqoveyhjujffe52_dir
# I ran cargo clean so I had enough disk space
$ rustup component remove rust-docs
info: removing component 'rust-docs'
warning: during uninstall component rust-docs was not found
$ ls ~/.local/lib/rustup/toolchains/stable-x86_64-unknown-linux-gnu/share/
doc  man  zsh
$ rustup toolchain install stable
info: syncing channel updates for 'stable-x86_64-unknown-linux-gnu'

  stable-x86_64-unknown-linux-gnu unchanged - rustc 1.47.0 (18bf6b4f0 2020-10-07)

error: rustup is not installed at '/acct/jynelson/.local/lib/cargo'

Notes

Output of rustup --version: rustup 1.20.2 (13979c968 2019-10-16)
Output of rustup show:

Default host: x86_64-unknown-linux-gnu
rustup home:  /acct/jynelson/.local/lib/rustup

I'd test with a newer version, but apparently rustup self update is also confused:

$ rustup self update
error: rustup is not installed at '/acct/jynelson/.local/lib/cargo'

Start by reproducing the failure with rustup component remove rust-docs, then compare the subsequent rustup component remove rust-docs, rustup toolchain install stable, and rustup self update behavior. Trace the component-removal path around the temp-directory error and determine what tests cover the resulting state; done means the failure is handled consistently without leaving the installation confused.