This would hide the installation process, but the toolchain will still be installed.
What the UX would be for the case when a toolchain for the current path is missing?
Installing the toolchain without any visual indication seems like a degradation, not an improvement.
And, I think, the prompt will also just hang until the installation is finished.

Yah, the prompt won't update until the download is finished, and even worse, the download will be restarted every time you press enter, because Tide kills the background shell that is doing the downloading.

Here's what Starship does with this same problem: https://github.com/starship/starship/blob/master/src/modules/rust.rs#L51-L89

// $CARGO_HOME/bin/rustc(.exe) --version may attempt installing a rustup toolchain.
// starship/starship#417
//
// To display appropriate versions preventing rustc from downloading toolchains, we have to
// check
// 1. $RUSTUP_TOOLCHAIN
// 2. The override list from ~/.rustup/settings.toml (like rustup override list)
// 3. rust-toolchain or rust-toolchain.toml in . or parent directories
// 4. The default_toolchain from ~/.rustup/settings.toml (like rustup default)
// 5. rustup default (in addition to the above, this also looks at global fallback config files)
// as rustup does.
// https://github.com/rust-lang/rustup.rs/tree/eb694fcada7becc5d9d160bf7c623abe84f8971d#override-precedence
//
// Probably we have no other way to know whether any toolchain override is specified for the
// current directory. The following commands also cause toolchain installations.
// - rustup show
// - rustup show active-toolchain
// - rustup which

We should probably copy their logic. I'd be happy to accept PRs, but I probably won't be working on this myself in the near term.

For now, it's better to expose the bad behavior to the user than to cover it up.