fix(coderd/x/chatd): resolve inflight race (#26460)

Using `WaitGroup.Go` must be synchronized with `WaitGroup.Wait`
according to [go docs](https://pkg.go.dev/sync#WaitGroup.Go):

> If the WaitGroup is empty, Go must happen before a
[WaitGroup.Wait](https://pkg.go.dev/sync#WaitGroup.Wait).

There were a couple of places in chatd that violated this principle.
This was caught as a data race in
https://github.com/coder/internal/issues/1599. This PR ensures that all
functions that spawn inflight goroutines synchronize with each other.

I also noticed that inflight goroutines may be spawned after the server
is closed, which was surprising and looked like a bug. This PR therefore
also introduces a mechanism that disallows spawning inflight goroutines
after the server is closed, and ensures that any code that tries doing
it logs an error.

Closes https://github.com/coder/internal/issues/1599.

coderd/x/chatd/quickgen.go

@@ -169,7 +169,7 @@ func (p *Server) GenerateChatTitleAsync(ctx context.Context, chat database.Chat)
 	// Detach from the request lifetime so title generation can finish
 	// even after the create response is written.
 	titleCtx := context.WithoutCancel(ctx)
-	p.inflight.Go(func() {
+	if err := p.goInflight(func() {
 		modelOpts := modelBuildOptionsFromMessages(messages)
 		titleCtx = withActiveTurnAPIKeyID(titleCtx, modelOpts)
 		model, modelConfig, keys, route, _, _, _, err := p.resolveChatModel(titleCtx, chat, modelOpts)
@@ -193,7 +193,13 @@ func (p *Server) GenerateChatTitleAsync(ctx context.Context, chat database.Chat)
 			logger,
 			p.existingDebugService(),
 		)
-	})
+	}); err != nil {
+		logger.Error(titleCtx, "failed to schedule automatic chat title generation",
+			slog.F("chat_id", chat.ID),
+			slog.F("owner_id", chat.OwnerID),
+			slog.Error(err),
+		)
+	}
 }
 
 // maybeGenerateChatTitle generates an AI title for the chat when