I just noticed that the submit button returns false onclientclick. can you remove that and see if that does anything? I'm not sure why the template does this, let me see what I can find out...
However, If you only wish to style the box it isn't necessarily required that you remap an external template. the submit button uses the class .sfsearchSubmit so you can apply classes to that directly in your stylesheet.
Here is a similar style I used in an example a while back to show an image instead of the standard search box. no external template was used:
notice I cheated and used 'transparent' for the font color to hide it. there are other strageties you could use (text-indent, font-size, jQuery, etc) to hide the text in the search box.
if you wish to actually clear it I believe then you would need the template....
I hope this is helpful!