The reason rust-analyzer doesn't update diagnostics until you save is historical. Originally, people tried to build IDE support by reusing rustc itself, but this proved too slow and cumbersome at the time.
Rust-analyzer reimplemented the frontend in a more IDE-friendly architecture, but focused more on name resolution than on type checking. So it delegated diagnostics to literally just running `cargo check`.
As parts of rustc get rewritten over time (the trait solver, borrow checker) they have also been made more IDE-friendly and reusable, so rust-analyzer is slowly gaining the ability to surface more type checking diagnostics as you edit, without delegating to `cargo check`.
Rust-analyzer reimplemented the frontend in a more IDE-friendly architecture, but focused more on name resolution than on type checking. So it delegated diagnostics to literally just running `cargo check`.
As parts of rustc get rewritten over time (the trait solver, borrow checker) they have also been made more IDE-friendly and reusable, so rust-analyzer is slowly gaining the ability to surface more type checking diagnostics as you edit, without delegating to `cargo check`.