If you're using VS Code then you can add `"rust-analyzer.check.command": "clippy"` to your `settings.json`. I assume there's a similar setting for rust-analyzer in other editors.
Package managers are for running other people's code, I would not expect the same of static analysis tools, especially since they are of use while auditing other people's code before building/running it.
Cargo's threat model here is identical to that of rust analyser. If you trust your dependency tree sufficiently to run `cargo build`, then you trust it sufficiently to run rust analyser.
Rust analyser executes those exact same build scripts. This is the primary avenue for exploits in both pieces of software, and is called out explicitly in the page you originally linked.