Additional info I came across while becoming even more exasperated:
1. The white gap is 48 pixels high.
2. This only happens on the page that is set as the front page of the site. If I pick a different page to be the static front page, and then I view the page that used to be the front page (and was having the issue) there is now magically no gap between the black title bar and the featured image.
I really hope someone knows how to fix this. I have spent hours trying to fix this, when I should have been done with the whole site by now.