Usando LogMetric em um modo isolado do Azure Function 8.0

O problema parece estar nos seus pacotes e depois de alterar os pacotes abaixo, funcionou para mim:

Eu usei Microsoft.Azure.Functions.Worker.ApplicationInsightso pacote.

Meus pacotes no csproj:

  <ItemGroup>
    <FrameworkReference Include="Microsoft.AspNetCore.App" />
    <PackageReference Include="Microsoft.Azure.Functions.Worker" Version="1.21.0" />
    <PackageReference Include="Microsoft.Azure.Functions.Worker.Extensions.Http" Version="3.1.0" />
    <PackageReference Include="Microsoft.Azure.Functions.Worker.Extensions.Http.AspNetCore" Version="1.2.1" />
    <PackageReference Include="Microsoft.Azure.Functions.Worker.Sdk" Version="1.17.0" />
    <PackageReference Include="Microsoft.ApplicationInsights.WorkerService" Version="2.22.0" />
    <PackageReference Include="Microsoft.Azure.Functions.Worker.ApplicationInsights" Version="1.2.0" />
    <PackageReference Include="Microsoft.Extensions.Configuration.UserSecrets" Version="8.0.0" />
  </ItemGroup>
  

Função1.cs:

using Microsoft.AspNetCore.Http;
using Microsoft.AspNetCore.Mvc;
using Microsoft.Azure.Functions.Worker;
using Microsoft.Extensions.Logging;

namespace FunctionApp182
{
    public class Function1
    {
        private readonly ILogger<Function1> _logger;
        public Function1(ILogger<Function1> logger)
        {
            _logger = logger;
        }
        [Function("Function1")]
        public IActionResult Run([HttpTrigger(AuthorizationLevel.Function, "get", "post")] HttpRequest req)
        {
            _logger.LogInformation("Hello Rithwik, Func Started");
            _logger.LogMetric("rithwik", 8);
            return new OkObjectResult("Hello Rithwik, Func Ended");
        }
    }
}

Programa.cs:

using Microsoft.Azure.Functions.Worker;
using Microsoft.Extensions.DependencyInjection;
using Microsoft.Extensions.Hosting;

var host = new HostBuilder()
    .ConfigureFunctionsWebApplication()
    .ConfigureServices(services =>
    {
        services.AddApplicationInsightsTelemetryWorkerService();
        services.ConfigureFunctionsApplicationInsights();
    })
    .Build();

host.Run();

Saída:

No Application Insights:

Em locais: