Test that pixad writes only its own errors to stderr (closes #359)

Both tests run pixad as a child process. They fail against this code:
every start writes the dependency-injection library's event log to
stderr, and a startup error is printed twice inside its frames.

Model: opus-5-5

internal/app/app.go

@@ -2,13 +2,16 @@
 package app
 
 import (
+	"context"
 	"fmt"
+	"io"
 	"os"
 	"os/signal"
 	"syscall"
 
 	"github.com/prometheus/client_golang/prometheus"
 	"github.com/spf13/cobra"
+	"go.uber.org/dig"
 	"go.uber.org/fx"
 	"sneak.berlin/go/pixa/internal/config"
 	"sneak.berlin/go/pixa/internal/database"
@@ -53,7 +56,10 @@ func run(_ *cobra.Command, _ []string) {
 	// A write to a closed stdout or stderr must not end the process.
 	signal.Ignore(syscall.SIGPIPE)
 
-	fx.New(
+	app := fx.New(
+		// pixad logs with slog only; runApp prints a startup or shutdown
+		// error itself.
+		fx.NopLogger,
 		fx.Provide(
 			config.New,
 			database.New,
@@ -71,5 +77,40 @@ func run(_ *cobra.Command, _ []string) {
 			func(log *logger.Logger) { log.Identify() },
 			func(*server.Server) {},
 		),
-	).Run()
+	)
+
+	exitCode := runApp(app, os.Stderr)
+	if exitCode != 0 {
+		os.Exit(exitCode)
+	}
+}
+
+// runApp starts app, waits for a signal or a shutdown request, stops app,
+// and returns the status pixad exits with, as fx.App.Run does. It prints
+// a start or stop error to stderr, once, as the error that caused it:
+// dig.RootCause drops the frames fx and dig wrap around it.
+func runApp(app *fx.App, stderr io.Writer) int {
+	startCtx, cancelStart := context.WithTimeout(context.Background(), app.StartTimeout())
+	defer cancelStart()
+
+	err := app.Start(startCtx)
+	if err != nil {
+		_, _ = fmt.Fprintln(stderr, dig.RootCause(err))
+
+		return 1
+	}
+
+	shutdown := <-app.Wait()
+
+	stopCtx, cancelStop := context.WithTimeout(context.Background(), app.StopTimeout())
+	defer cancelStop()
+
+	err = app.Stop(stopCtx)
+	if err != nil {
+		_, _ = fmt.Fprintln(stderr, dig.RootCause(err))
+
+		return 1
+	}
+
+	return shutdown.ExitCode
 }