Check out a few things:
- Do you have an ApplicationInsights.config file in the root of your file system (/CMS/ApplicationInsights.config).
- Do you have ALL the proper packages installed? Looks like you may be missing a few.
I'm running several Kentico v10.0.x sites in Azure and have 7 different ApplicationInsights packages installed to work properly. Also note, it takes a few minutes for things to start working in Azure. Last setup was around 10 minutes before I saw everything working properly.

ApplicationsInsights.config:
<?xml version="1.0" encoding="utf-8"?>
<ApplicationInsights xmlns="http://schemas.microsoft.com/ApplicationInsights/2013/Settings">
<TelemetryInitializers>
<Add Type="Microsoft.ApplicationInsights.DependencyCollector.HttpDependenciesParsingTelemetryInitializer, Microsoft.AI.DependencyCollector"/>
<Add Type="Microsoft.ApplicationInsights.WindowsServer.AzureRoleEnvironmentTelemetryInitializer, Microsoft.AI.WindowsServer"/>
<Add Type="Microsoft.ApplicationInsights.WindowsServer.AzureWebAppRoleEnvironmentTelemetryInitializer, Microsoft.AI.WindowsServer"/>
<Add Type="Microsoft.ApplicationInsights.WindowsServer.BuildInfoConfigComponentVersionTelemetryInitializer, Microsoft.AI.WindowsServer"/>
<Add Type="Microsoft.ApplicationInsights.Web.WebTestTelemetryInitializer, Microsoft.AI.Web"/>
<Add Type="Microsoft.ApplicationInsights.Web.SyntheticUserAgentTelemetryInitializer, Microsoft.AI.Web">
<Filters>search|spider|crawl|Bot|Monitor|AlwaysOn</Filters>
</Add>
<Add Type="Microsoft.ApplicationInsights.Web.ClientIpHeaderTelemetryInitializer, Microsoft.AI.Web"/>
<Add Type="Microsoft.ApplicationInsights.Web.OperationNameTelemetryInitializer, Microsoft.AI.Web"/>
<Add Type="Microsoft.ApplicationInsights.Web.OperationCorrelationTelemetryInitializer, Microsoft.AI.Web"/>
<Add Type="Microsoft.ApplicationInsights.Web.UserTelemetryInitializer, Microsoft.AI.Web"/>
<Add Type="Microsoft.ApplicationInsights.Web.AuthenticatedUserIdTelemetryInitializer, Microsoft.AI.Web"/>
<Add Type="Microsoft.ApplicationInsights.Web.AccountIdTelemetryInitializer, Microsoft.AI.Web"/>
<Add Type="Microsoft.ApplicationInsights.Web.SessionTelemetryInitializer, Microsoft.AI.Web"/>
</TelemetryInitializers>
<TelemetryModules>
<Add Type="Microsoft.ApplicationInsights.DependencyCollector.DependencyTrackingTelemetryModule, Microsoft.AI.DependencyCollector">
<ExcludeComponentCorrelationHttpHeadersOnDomains>
<Add>core.windows.net</Add>
<Add>core.chinacloudapi.cn</Add>
<Add>core.cloudapi.de</Add>
<Add>core.usgovcloudapi.net</Add>
</ExcludeComponentCorrelationHttpHeadersOnDomains>
</Add>
<Add Type="Microsoft.ApplicationInsights.Extensibility.PerfCounterCollector.PerformanceCollectorModule, Microsoft.AI.PerfCounterCollector">
</Add>
<Add Type="Microsoft.ApplicationInsights.Extensibility.PerfCounterCollector.QuickPulse.QuickPulseTelemetryModule, Microsoft.AI.PerfCounterCollector"/>
<Add Type="Microsoft.ApplicationInsights.WindowsServer.DeveloperModeWithDebuggerAttachedTelemetryModule, Microsoft.AI.WindowsServer"/>
<Add Type="Microsoft.ApplicationInsights.WindowsServer.UnhandledExceptionTelemetryModule, Microsoft.AI.WindowsServer"/>
<Add Type="Microsoft.ApplicationInsights.WindowsServer.UnobservedExceptionTelemetryModule, Microsoft.AI.WindowsServer"/>
<Add Type="Microsoft.ApplicationInsights.Web.RequestTrackingTelemetryModule, Microsoft.AI.Web">
<Handlers>
<Add>System.Web.Handlers.TransferRequestHandler</Add>
<Add>Microsoft.VisualStudio.Web.PageInspector.Runtime.Tracing.RequestDataHttpHandler</Add>
<Add>System.Web.StaticFileHandler</Add>
<Add>System.Web.Handlers.AssemblyResourceLoader</Add>
<Add>System.Web.Optimization.BundleHandler</Add>
<Add>System.Web.Script.Services.ScriptHandlerFactory</Add>
<Add>System.Web.Handlers.TraceHandler</Add>
<Add>System.Web.Services.Discovery.DiscoveryRequestHandler</Add>
<Add>System.Web.HttpDebugHandler</Add>
</Handlers>
</Add>
<Add Type="Microsoft.ApplicationInsights.Web.ExceptionTrackingTelemetryModule, Microsoft.AI.Web"/>
</TelemetryModules>
<TelemetryProcessors>
<Add Type="Microsoft.ApplicationInsights.Extensibility.PerfCounterCollector.QuickPulse.QuickPulseTelemetryProcessor, Microsoft.AI.PerfCounterCollector"/>
<Add Type="Microsoft.ApplicationInsights.WindowsServer.TelemetryChannel.AdaptiveSamplingTelemetryProcessor, Microsoft.AI.ServerTelemetryChannel">
<MaxTelemetryItemsPerSecond>5</MaxTelemetryItemsPerSecond>
</Add>
</TelemetryProcessors>
<TelemetryChannel Type="Microsoft.ApplicationInsights.WindowsServer.TelemetryChannel.ServerTelemetryChannel, Microsoft.AI.ServerTelemetryChannel"/>
</ApplicationInsights>