The 'AllowedValues' card is used when the data source itself knows what are all the possible values (for example, an enumeration column in CDM). You can, however, customize the card to change the text input control to another more suitable for your needs, by unlocking the card.
Let's walk through it... I'll create an app from data from a SQL table that has a property that I want to restrict the values. Here's the edit screen for it:

You can notice that the card has a small "padlock" indicating that it's locked and cannot be changed. With the card selected, click the '...' button and then click 'Advanced options':

Now click the padlock to unlock the card, to be able to change the controls used on it:

You'll see that the padlock on the card name disappeared. Select the control that you want to delete (the text input). You'll notice that the advanced pane shows the control name (in this case, 'DataCardValue7') - take a note of this name as it will make it easier to correct some errors later.

Now delete the control. You'll notice that there will be some errors in the card, because that control was referenced by others.

Now add a new control (Insert -> Dropdown) while the card is selected (so that the control will be added as a child of the card. After adding it, click on the control name, and rename it to the name of the original control (in this example, 'DataCardValue7')

Now go back to the advanced pane and update the 'Items' property to the list of values that you want to allow. Also update the 'Default' value to 'Parent.Default' so that it will show the current value for the record when it's being edited.

Finally, select the card, and change its Update property to get the value from the dropdown. In this case, it's set to 'DataCardValue7.Selected.Value'

And now your edit screen only shows the items from the "allowed values" for your scenario.