"It just doesn't make sense!"
<form action="process_page.php" id="process_form" method="post"> <!-- form inputs here --> <input name="action" type="hidden" value="add" /> <!-- main problem --></form>
The jQuery code is the proper syntax to achieve this. But the DOM has been setup wrong. Notice there was an input with the name "action" which actually overrides the form attribute "action".
Rename your input field name attribute. If it still isn't working also check the id attribute name too.