I am sorry to know you are facing such inconsistent behavior.
First thing to verify is what Visual Studio you are using - is this Professional or Enterprise? If you are using the Community edition - please note that this is not supported.
Though, if you the Visual Studio 2019 is minimum Professional and has the latest updates installed, I would suggest we verify the Test Studio installation wizard first. Please go through the below steps and let me know the outcome.
1. Close any Test Studio and Visual Studio instances.
2. Open the Windows Control Panel and open the Test Studio installation to modify it.
3. Go to the Change view and ensure the Visual Studio 2019 Plugin components are selected.
4. Click the Back button to get to the initial screen and choose to Repair the installation.
5. Once ready, open the project in Visual Studio 2019 and check if the Telerik menu is available.
6. If still not, please uninstall Test Studio from the Windows control panel completely and then perform a new fresh installation, but with generating the installer log.
Please, if still there is no change, send that log zipped for further analysis.
Thank you in advance for your time and cooperation.