Auto instrumention - Metrics work, traces not received

The failure to set up an ETW trace session should not be an issue for agent startup, but it does mean you wouldn't get GC metrics. You can disable GC metrics with

<add key="ElasticApm:DisableMetrics" value="clr.gc.*" />

which won't attempt to create an ETW trace session.

From the APM agent log posted, it looks like there is still an attempt to instantiate the singleton instance of the agent when it has already been instantiated

[2022-02-23 10:06:28.649 +01:00][Error] - The singleton APM agent has already been instantiated and can no longer be configured. Reusing existing instance
[2022-02-23 10:06:28.649 +01:00][Trace] - Initialization - Agent.Setup called. Callstack:    at Elastic.Apm.Agent.Setup(AgentComponents agentComponents)
   at Elastic.Apm.AspNetFullFramework.ElasticApmModule.<>c__DisplayClass25_0.<InitOnceForAllInstancesUnderLock>b__0()
   at Elastic.Apm.Helpers.LazyContextualInit.<>c__DisplayClass6_0.<Init>b__0()
   at System.Threading.LazyInitializer.EnsureInitializedCore[T](T& target, Boolean& initialized, Object& syncLock, Func`1 valueFactory)
   at Elastic.Apm.Helpers.LazyContextualInit.Init(Action initAction)
   at Elastic.Apm.AspNetFullFramework.ElasticApmModule.InitOnceForAllInstancesUnderLock(String dbgInstanceName)
   at Elastic.Apm.AspNetFullFramework.ElasticApmModule.InitImpl(HttpApplication application)
   at Elastic.Apm.AspNetFullFramework.ElasticApmModule.Init(HttpApplication application)
   at System.Web.HttpApplication.RegisterEventSubscriptionsWithIIS(IntPtr appContext, HttpContext context, MethodInfo[] handlers)
   at System.Web.HttpApplication.InitSpecial(HttpApplicationState state, MethodInfo[] handlers, IntPtr appContext, HttpContext context)
   at System.Web.HttpApplicationFactory.GetSpecialApplicationInstance(IntPtr appContext, HttpContext context)
   at System.Web.Hosting.PipelineRuntime.InitializeApplication(IntPtr appContext)

I'm fairly sure this is the crux of why we're not seeing traces.

Something worth trying is to disable the ASP.NET integration in the profiler, and add the ElasticApmModule manually in web.config, by referencing the NuGet Gallery | Elastic.Apm.AspNetFullFramework 1.14.0 package. The integration can be disabled with the environment variable

ELASTIC_APM_PROFILER_EXCLUDE_INTEGRATIONS=AspNet

added to the app pool. You'll need to recycle the app pool for changes to take effect.