Add process-scoped SQLite telemetry (#22154)

## Summary
- add SQLite init, backfill-gate, and fallback telemetry without
introducing a cross-cutting state-db access wrapper
- install one process-scoped telemetry sink after OTEL startup and let
low-level state/rollout paths emit through it directly
- add process-start metrics for the process owners that initialize
SQLite

---------

Co-authored-by: Owen Lin <owen@openai.com>

codex-rs/core/src/otel_init.rs

@@ -99,3 +99,18 @@ pub fn build_provider(
 pub fn codex_export_filter(meta: &tracing::Metadata<'_>) -> bool {
     meta.target().starts_with("codex_otel")
 }
+
+pub fn record_process_start(otel: Option<&OtelProvider>, originator: &str) {
+    let Some(metrics) = otel.and_then(OtelProvider::metrics) else {
+        return;
+    };
+    let _ = codex_otel::record_process_start_once(metrics, originator);
+}
+
+pub fn install_sqlite_telemetry(otel: Option<&OtelProvider>, originator: &str) {
+    let Some(metrics) = otel.and_then(OtelProvider::metrics) else {
+        return;
+    };
+    let telemetry = codex_rollout::sqlite_telemetry_recorder(metrics.clone(), originator);
+    let _ = codex_state::install_process_db_telemetry(telemetry);
+}