[Router] Log every request at one site; derive its outcome from the final status (3/3) (#39465)

Co-authored-by: Kangyan Zhou <kangyan.zhou@radixark.ai>
Co-authored-by: Claude Opus 5 (1M context) <noreply@anthropic.com>
Co-authored-by: Shangming Cai <csmthu@gmail.com>

experimental/sgl-router/Cargo.toml

@@ -49,7 +49,7 @@ axum = { version = "0.8", features = ["macros", "tracing", "http2"] }
 # `Frame`/`SizeHint` are not re-exported through `axum::body`.
 http-body = "1"
 tower = { version = "0.5", features = ["full"] }
-tower-http = { version = "0.6", features = ["trace", "compression-gzip", "cors", "timeout", "request-id"] }
+tower-http = { version = "0.6", features = ["trace", "compression-gzip", "cors", "timeout", "request-id", "catch-panic"] }
 reqwest = { version = "0.12", features = ["stream", "json", "rustls-tls", "http2"], default-features = false }
 sgl-kv-indexer = { path = "sgl-kv-indexer" }
 
@@ -104,6 +104,9 @@ http-body-util = "0.1"
 # rather than as an unresolved import in a test.
 hyper = { version = "1", features = ["server", "http1", "http2"] }
 hyper-util = { version = "0.1", features = ["tokio"] }
+# Capture the router's `tracing` access log in tests. Also a runtime dep, but
+# integration tests are a separate crate and need it declared here too.
+tracing-subscriber = { version = "0.3", features = ["fmt"] }
 serde = { version = "1", features = ["derive"] }
 serde_json = "1"
 tempfile = "3"