First I'd like to say that if you use ContentTemplate the other code that you've put in the content won't be applied. That's why nothing happens on "Add Patient" button click - in the event handler you're hiding/showing the grids from the content when you need the grids from the contentTemplate.
If you're going to define all your RadTileViewItems in XAML I think you don't need contentTemplate - you could just add everything in the content, but if you're going to use templates you'll have to change your code a little bit. The best approach will be to use command in your ViewModel and bind the visibility of your parts (the grid and the textblock) to some properties, but a possible workaround is to use the parentOfType and childrenOfType methods in an event handler to do the same in code-behind.
I've changed your project a little bit and attached it so could you please examine it and if you have further questions feel free to ask.
the Telerik team