Avoid severity-level-0 logging in application insights from function app

Take a look at the different log categories here.

I'd say to reduce logging but keep important information like requests and dependencies set the Microsoft, Worker and Function.<YOUR_FUNCTION_NAME> categories to LogLevel warning (or none), but keep Function.<YOUR_FUNCTION_NAME>.User and all the others at information. This will remove most logging but keep statistics and the success/failure logs available in the portal.

By doing loggingBuilder.AddApplicationInsights(key).SetMinimumLevel(LogLevel.Warning); you are only changing the loglevel for your the Function category, not the others.

{
  "version": "2.0",
  "logging": {
    "applicationInsights": {
      "samplingSettings": {
        "isEnabled": true,
        "excludedTypes": "Request"
      }
    },
    "logLevel": {
      "Function.<YOUR_FUNCTION_NAME>": "Warning",
      "Function.<YOUR_FUNCTION_NAME>.User": "Information",
      "Microsoft": "Warning",
      "Worker": "Warning",
      "default": "Information"
    }
  }
}

The category Function.<YOUR_FUNCTION_NAME> is a tricky one. It generates both useful logs and some noise. Try both Warning and Information as log level and see what best works for you.

EDIT The way you configure the function is wrong. Application Insights is added by Azure Functions automatically (source). If you want to directly use a TelemetryClient you need to add the NuGet package Microsoft.Azure.WebJobs.Logging.ApplicationInsights and that is it.

The default integration will pick up the configuration from the host.json file. Right now you are setting up the logging manually which will ignore the settings since the configuration is not loaded. That is why, for example, the message Poll for function '{name of function in fa1}' on queue '{...}' with ClientRequestId '{...}' found 0 messages in 5 ms. that belongs to category Microsoft.Azure.WebJobs.Host.Queues.Listeners.QueueListener is logged and not filtered. So remove the code you have shown in the question.