The simplest way is to get rid of:
<<remove "#introForm" >>
And change the button to:
<<button "Start!" `passage()`>>
That code will cause the passage to reload itself, which also stores the values of the variables so that if you reload the page, the form will remain hidden. Since $formIntroSubmitted will be true at that point, it will also hide the form and the "Start!" button.
That will cause a transition when you click the button, but hopefully that isn't a problem.
Also, you don't actually need the "<form>" elements, because it's not really a form.
NOTE: The marks around "passage()" are backquotes/backticks, found on the key with the tilde "~" on it. This causes it to pass the return value of that function, which is the name of the current passage, instead of passing the function itself. If the passage's name was, for example, "Form" you could do "<<button "Start!" "Form">>" instead.
Hope that helps!