Use rust-analyzer's flycheck as source of cargo diagnostics (#29779)

Follow-up of https://github.com/zed-industries/zed/pull/29706

Instead of doing `cargo check` manually, use rust-analyzer's flycheck:
at the cost of more sophisticated check command configuration, we keep
much less code in Zed, and get a proper progress report.

User-facing UI does not change except `diagnostics_fetch_command` and
`env` settings removed from the diagnostics settings.

Release Notes:

- N/A

assets/settings/default.json

@@ -935,22 +935,9 @@
       "max_severity": null
     },
     "rust": {
-      // When enabled, Zed runs `cargo check --message-format=json`-based commands and
-      // collect cargo diagnostics instead of rust-analyzer.
-      "fetch_cargo_diagnostics": false,
-      // A command override for fetching the cargo diagnostics.
-      // First argument is the command, followed by the arguments.
-      "diagnostics_fetch_command": [
-        "cargo",
-        "check",
-        "--quiet",
-        "--workspace",
-        "--message-format=json",
-        "--all-targets",
-        "--keep-going"
-      ],
-      // Extra environment variables to pass to the diagnostics fetch command.
-      "env": {}
+      // When enabled, Zed disables rust-analyzer's check on save and starts to query
+      // Cargo diagnostics separately.
+      "fetch_cargo_diagnostics": false
     }
   },
   // Files or globs of files that will be excluded by Zed entirely. They will be skipped during file