Otelhttp metrics are not exporter to prometheus with meter option

[kirill-lokhmatov]

Description

Environment

• OS: linux
• Architecture: x64
• Go Version: 1.21
• otelhttp version: v0.52.0

Steps To Reproduce

`

   prometheusExporter, err := prometheus.New(
	prometheus.WithoutTargetInfo(),
	prometheus.WithoutScopeInfo())
if err != nil {
	return nil, err
}
provider := metric.NewMeterProvider(
	metric.WithReader(prometheusExporter),
)
loggedHttpTransport := transport.NewLoggedHttpTransport(logger)
client := &http.Client{
	Transport: otelhttp.NewTransport(otelhttp.WithMeterProvider(provider))}```

No metrics in prom (i am doing the same when working with otelsql and see the metrics)

Expected behavior

Http client metrics are exported to prometheus (like in otelsql)

[dmathieu]

Could you provide us with actually runnable steps to reproduce?
I have tried reproducing with an adaptation from the prometheus example, and I am properly getting metrics.

// Copyright The OpenTelemetry Authors
// SPDX-License-Identifier: Apache-2.0

package main

import (
	"io"
	"log"
	"net/http"

	"github.com/prometheus/client_golang/prometheus/promhttp"

	"go.opentelemetry.io/contrib/instrumentation/net/http/otelhttp"
	"go.opentelemetry.io/otel/exporters/prometheus"
	"go.opentelemetry.io/otel/sdk/metric"
)

func main() {
	exporter, err := prometheus.New()
	if err != nil {
		log.Fatal(err)
	}
	provider := metric.NewMeterProvider(metric.WithReader(exporter))

	helloHandler := func(w http.ResponseWriter, req *http.Request) {
		_, _ = io.WriteString(w, "Hello, world!\n")
	}

	otelHandler := otelhttp.NewHandler(
		http.HandlerFunc(helloHandler),
		"Hello",
		otelhttp.WithMeterProvider(provider),
	)

	http.Handle("/metrics", promhttp.Handler())
	http.Handle("/hello", otelHandler)
	err = http.ListenAndServe(":7777", nil)
	if err != nil {
		log.Fatal(err)
	}
}