How to delete values in Link field?
It's perhaps not that obvious but all you need to do is empty the text field containing the value you want to remove. Once you save the entity the value will be gone.

Files need a remove button as there's no other way to 'empty' that particular field (a round-trip to the server is required to purge the file that was temporarily uploaded). Other, less complex fields (like text fields), don't need to contact the server via Ajax to do that, so I guess the remove button is deemed unnecessary. 

If you're interested in the mechanics of this, have a look at the hook_field_is_empty function, which all field modules are required to implement.