If you experiment weird behavior of template switcher and the emulator shows your standard desktop template probably you have some extensions installed that block Responsivizer.
Often some security restrictions, plugins, or extensions aimed to enforce Joomla security block some url variables or parameters.
One of the most powerful and known extension is Admin Tools from Akeeba team. If you have Admin Tools installed on your Joomla site, check the image below to ensure that the 'template' parameter is not blocked. This usually solves every kind of issues.