Cargo `is not applicable` when using a toolchain specified in `rust-toolchain.toml`

Problem

I'm experiencing an issue very similar to https://github.com/rust-lang/rustup/issues/3025

In Nushell, we currently use this in our rust-toolchain.toml:

[toolchain]
profile = "default"
channel = "1.64.0"

On my mac, this is now broken:

❯ cargo install --path . --features=extra
error: the 'cargo' binary, normally provided by the 'cargo' component, is not applicable to the '1.64.0-aarch64-apple-darwin' toolchain

Strangely enough, if I change channel = "1.64.0" to channel = "1.64" it fixes the issue but breaks other platforms. It almost seems like there's a naming issue that got published.

Steps

1. Described above

Possible Solution(s)

It should be possible to use rust-toolchain.toml to request an earlier version of the Rust compiler using the full version number across platforms.

Notes

No response

Rustup version

rustup 1.25.1 (bb60b1e89 2022-07-12)
info: This is the version for the rustup toolchain manager, not the rustc compiler.
info: The currently active `rustc` version is `rustc 1.64.0 (a55dd71d5 2022-09-19)`

Installed toolchains

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

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

stable-aarch64-apple-darwin (default)
nightly-2022-01-20-aarch64-apple-darwin
nightly-2022-10-20-aarch64-apple-darwin
nightly-aarch64-apple-darwin
1.64-aarch64-apple-darwin
1.56.0-aarch64-apple-darwin
1.61.0-aarch64-apple-darwin
1.63.0-aarch64-apple-darwin
1.64.0-aarch64-apple-darwin

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

1.64.0-aarch64-apple-darwin (overridden by '/Users/jt/Source/nushell/rust-toolchain.toml')
rustc 1.64.0 (a55dd71d5 2022-09-19)

Reproduce the issue with the provided rust-toolchain.toml on an aarch64-apple-darwin host, using channel "1.64.0" and the cargo install --path . --features=extra command. Compare it with channel "1.64" and inspect rustup's toolchain resolution and target naming; done means the full version works across platforms without making the cargo component inapplicable.