`rustup` installation broken after upgrading from macOS 13.6 to 14.2

Problem

I installed Rust using rustup some time last year (or maybe earlier, I forget) onto a Macbook Air M1 running macOS 13. A few weeks ago I upgraded from 13.6 to 14.2, and since then my Rust installation has been subtly broken in ways I keep discovering.

Steps

1. Install Rust via rustup on macOS 13.6.
2. Upgrade to macOS 14.2.
3. Try to do things that require interacting with environment variables.

Possible Solution(s)

No response

Notes

I first noticed problems when I started trying to cross-compile for iOS targets. See https://github.com/rust-lang/rust/issues/114276#issuecomment-1911446978 for more information (as that issue seems to be the same as mine, so I posted a comment there initially).

I just now noticed that the rustup overrides are not working (neither manual not rust-toolchain.toml), which is why I now think it's a rustup problem rather than a rustc or cargo problem:

$ rustup --version
rustup 1.26.0 (5af9b9484 2023-04-05)
info: This is the version for the rustup toolchain manager, not the rustc compiler.
info: The currently active `rustc` version is `rustc 1.75.0 (82e1608df 2023-12-21)`

$ cargo --version
cargo 1.75.0

$ cd rage/
$ rustup --version
rustup 1.26.0 (5af9b9484 2023-04-05)
info: This is the version for the rustup toolchain manager, not the rustc compiler.
info: The currently active `rustc` version is `rustc 1.65.0 (897e37553 2022-11-02)`

$ cargo --version
cargo 1.75.0

Rustup version

rustup 1.26.0 (5af9b9484 2023-04-05)

Installed toolchains

rustup show
Default host: aarch64-apple-darwin
rustup home:  /Users/str4d/.rustup

installed toolchains
--------------------

stable-aarch64-apple-darwin (default)
beta-aarch64-apple-darwin
nightly-2022-11-03-aarch64-apple-darwin
nightly-2022-12-26-aarch64-apple-darwin
nightly-2023-03-10-aarch64-apple-darwin
nightly-2023-05-03-aarch64-apple-darwin
nightly-2023-06-09-aarch64-apple-darwin
nightly-aarch64-apple-darwin
1.56.0-aarch64-apple-darwin
1.56.1-aarch64-apple-darwin
1.59.0-aarch64-apple-darwin
1.60.0-aarch64-apple-darwin
1.65.0-aarch64-apple-darwin
1.67.1-aarch64-apple-darwin
1.69.0-aarch64-apple-darwin

active toolchain
----------------

1.65.0-aarch64-apple-darwin (overridden by '/Users/str4d/dev/rust/age/rage/rust-toolchain.toml')
rustc 1.65.0 (897e37553 2022-11-02)

Start by reproducing the reported upgrade from macOS 13.6 to 14.2 on an Apple Silicon Mac, focusing on environment-variable handling and the manual and rust-toolchain.toml overrides. Compare the behavior with the linked Rust issue and the supplied rustup, cargo, and toolchain outputs; done means overrides and related environment-variable interactions work after the upgrade.