When setting the header color in the customizer (after selecting manage background by color), the changes show in the preview pane but when the changes are published it does not reflect on the site. Usually this is a cache issue in my browser but I’ve tried multiple browsers from multiple devices. The header on the home page stays transparent while the rest of the pages still show the default wood grain background.
No matter what you select in the customizer for the header background the mobile version always goes to white and if you have a log that is white it disappears. This can be seen on your demo page if you resize your browser window.