fix(server): preserve LLM error source chain in doctor output

`fabro_llm::Error`'s Display only renders the top-level message field for
`Network`/`Stream`/`Configuration`/`RequestTimeout` variants — the
`#[source]` chain is dropped. Walk the chain at the rendering boundary
so connectivity failures (DNS, connection refused, TLS) surface their
underlying cause in `fabro doctor` output.

Per docs/internal/error-handling-strategy.md, CLI surfaces should render
the full cause chain. Adds a regression test that walks `err.source()`
on a typed Network error with an inner io::Error.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

lib/crates/fabro-server/src/diagnostics.rs

@@ -121,11 +121,12 @@ async fn check_llm_providers(state: &AppState) -> CheckResult {
         match probe_result {
             Ok(Ok(())) => details.push(CheckDetail::new(format!("{provider}: OK"))),
             Ok(Err(err)) => {
+                let rendered = render_error_chain(&err);
                 failures.push(ProviderFailure {
                     name:  provider.to_string(),
-                    short: short_error_line(&err.to_string()),
+                    short: short_error_line(&rendered),
                 });
-                details.push(CheckDetail::new(format!("{provider}: {err}")));
+                details.push(CheckDetail::new(format!("{provider}: {rendered}")));
             }
             Err(_) => {
                 failures.push(ProviderFailure {
@@ -188,6 +189,17 @@ fn short_error_line(rendered: &str) -> String {
     }
 }
 
+fn render_error_chain(err: &dyn std::error::Error) -> String {
+    let mut out = err.to_string();
+    let mut source = err.source();
+    while let Some(cause) = source {
+        out.push_str(": ");
+        out.push_str(&cause.to_string());
+        source = cause.source();
+    }
+    out
+}
+
 fn probe_model(provider: Provider) -> String {
     Catalog::builtin()
         .probe_for_provider(provider)
@@ -675,6 +687,23 @@ mod tests {
         assert_eq!(short_error_line(&input), expected);
     }
 
+    #[test]
+    fn render_error_chain_includes_underlying_cause_for_typed_llm_error() {
+        let inner = std::io::Error::new(std::io::ErrorKind::ConnectionRefused, "tcp connect");
+        let err = fabro_llm::Error::network("openai request failed", inner);
+
+        let rendered = render_error_chain(&err);
+
+        assert!(
+            rendered.contains("openai request failed"),
+            "top-level message missing: {rendered}"
+        );
+        assert!(
+            rendered.contains("tcp connect"),
+            "underlying cause missing — chain not walked: {rendered}"
+        );
+    }
+
     #[test]
     fn short_error_line_returns_first_non_empty_trimmed_line() {
         let input = "  first line  \nsecond line";