My ObservableCollection has three items, and the rows are consistently shown in the datagrid. I cannot turn into edit mode a single cell of my DataGrid. I tried click, click
You have placed TextBlock in cell template as well as in cell editing template. That's why you are not noticing any change on pressing F2 and double-clicking the cell since no matter what it will always be TextBlock which you can't edit.
Either placed TextBox in your CellEditingTemplate like this -
Or either simply use the DataGridTextColumn in place of DataGridTemplateColumn which internally provides the support what are you trying to achive by the above code -